◗✉❡st✐♦♥ ■✳ ■s t❤✐s ❝♦♥❥❡❝t✉r❡ tr✉❡ ❢♦r ❊ ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ❄ ❚❤❡ ❉❛❧❡s✕➏❡❧❛③❦♦ ❝♦♥❥❡❝t✉r❡ ❈♦♥❥❡❝t✉r❡ ✭❉❛❧❡s✕➏❡❧❛③❦♦ ✷✵✶✷✮✳ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡✈❡r② ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧✳ ❊✈✐❞❡♥❝❡✳ ❋❡rr❡✐r❛ ❛♥❞ ❚♦♠❛ss✐♥✐ ✭✶✾✼✽✮ ❤❛✈❡ ♣r♦✈❡❞ ❛ str♦♥❣❡r ❢♦r♠ ♦❢ t❤✐s ❝♦♥❥❡❝t✉r❡ ❢♦r A ❝♦♠♠✉t❛t✐✈❡✳ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ❝❛s❡✿ ♦♣❡♥✦ ✸
❚❤❡ ❉❛❧❡s✕➏❡❧❛③❦♦ ❝♦♥❥❡❝t✉r❡ ❈♦♥❥❡❝t✉r❡ ✭❉❛❧❡s✕➏❡❧❛③❦♦ ✷✵✶✷✮✳ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ s✉❝❤ t❤❛t ❡✈❡r② ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❤❡♥ A ✐s ✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧✳ ❊✈✐❞❡♥❝❡✳ ❋❡rr❡✐r❛ ❛♥❞ ❚♦♠❛ss✐♥✐ ✭✶✾✼✽✮ ❤❛✈❡ ♣r♦✈❡❞ ❛ str♦♥❣❡r ❢♦r♠ ♦❢ t❤✐s ❝♦♥❥❡❝t✉r❡ ❢♦r A ❝♦♠♠✉t❛t✐✈❡✳ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ ❝❛s❡✿ ♦♣❡♥✦ ◗✉❡st✐♦♥ ■✳ ■s t❤✐s ❝♦♥❥❡❝t✉r❡ tr✉❡ ❢♦r A = B ( ❊ ) ✱ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❛❧❧ ❜♦✉♥❞❡❞✱ ❧✐♥❡❛r ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ❄ ✸
❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ❊ ♥ ♥ ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ❊ ✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ① ♥ ✇✐t❤ ① ♥ s✉❝❤ t❤❛t ❊ ♥ ♥ ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ ① ① ♥ ♥ ✶ t❤❛t ✐s✱ t❤❡ ♥❡t ① ♥ ◆ ✜♥✐t❡ ♥ ◆ ❝♦♥✈❡r❣❡s t♦ ① ✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B ( ❊ ) ❝♦♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ ℵ ✵ ✳ ✹
❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ❊ ♥ ♥ ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ❊ ✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ① ♥ ✇✐t❤ ① ♥ s✉❝❤ t❤❛t ❊ ♥ ♥ ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ ① ① ♥ ♥ ✶ t❤❛t ✐s✱ t❤❡ ♥❡t ① ♥ ◆ ✜♥✐t❡ ♥ ◆ ❝♦♥✈❡r❣❡s t♦ ① ✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B ( ❊ ) ❝♦♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ✹
t❤❛t ✐s✱ t❤❡ ♥❡t ① ♥ ◆ ✜♥✐t❡ ♥ ◆ ❝♦♥✈❡r❣❡s t♦ ① ✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B ( ❊ ) ❝♦♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ( ❊ ♥ ) ♥ ∈ N ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ❊ ✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ( ① ♥ ) ✇✐t❤ ① ♥ ∈ ❊ ♥ ( ♥ ∈ N ) s✉❝❤ t❤❛t ∞ � ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ ① = ① ♥ ♥ = ✶ ✹
❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳ ❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B ( ❊ ) ❝♦♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ( ❊ ♥ ) ♥ ∈ N ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ❊ ✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ( ① ♥ ) ✇✐t❤ ① ♥ ∈ ❊ ♥ ( ♥ ∈ N ) s✉❝❤ t❤❛t ∞ � ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ ① = ① ♥ ♥ = ✶ t❤❛t ✐s✱ t❤❡ ♥❡t �� � ① ♥ ◆ ⊂ N ✜♥✐t❡ ♥ ∈ ◆ ❝♦♥✈❡r❣❡s t♦ ① ✳ ✹
❆ ♣❛rt✐❛❧ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✳ ❚❤❡♥ B ( ❊ ) ❝♦♥t❛✐♥s ✷ c ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ❜✉t ♦♥❧② c ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s✱ ✇❤❡r❡ c = ✷ ℵ ✵ ✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ❛ s❡q✉❡♥❝❡ ( ❊ ♥ ) ♥ ∈ N ♦❢ ♥♦♥✲③❡r♦✱ ❝❧♦s❡❞ s✉❜s♣❛❝❡s ♦❢ ❊ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ❊ ✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ s❡q✉❡♥❝❡ ( ① ♥ ) ✇✐t❤ ① ♥ ∈ ❊ ♥ ( ♥ ∈ N ) s✉❝❤ t❤❛t ∞ � ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✱ ① = ① ♥ ♥ = ✶ t❤❛t ✐s✱ t❤❡ ♥❡t �� � ① ♥ ◆ ⊂ N ✜♥✐t❡ ♥ ∈ ◆ ❝♦♥✈❡r❣❡s t♦ ① ✳ ❘❡♠❛r❦✳ ❚❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ♥♦♥✲s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✳ ✹
t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ ✵ ✶ ✶ ♣ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ✶ ♣ ❛♥❞ ❝ ✵ ♣ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ ✵ ✶ ✶ ♣ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ ❡①❛♠♣❧❡s✿ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ✶ ♣ ❛♥❞ ❝ ✵ ♣ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ ✵ ✶ ✶ ♣ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ; ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ ❡①❛♠♣❧❡s✿ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ ✵ ✶ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ; ◮ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
t❤❡ s♣❛❝❡ ❈ ❑ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ; ◮ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ✶ ♣ ✳ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ; ◮ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ◮ t❤❡ s♣❛❝❡ ❈ ( ❑ ) ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ; ◮ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ◮ t❤❡ s♣❛❝❡ ❈ ( ❑ ) ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ ◮ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ( ✶ < ♣ < ∞ ) ✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ ✺
❆♣♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ t❤❡♦r❡♠ ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✱ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s s❡♣❛r❛❜❧❡ ❛♥❞ ❤❛s ❛ ❝♦✉♥t❛❜❧❡✱ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❞❡❝♦♠♣♦s✐t✐♦♥✿ ◮ ❊ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s ✭t❤✐s ♠❡❛♥s t❤❛t ❡❛❝❤ ♦❢ t❤❡ s✉❜s♣❛❝❡s ❊ ♥ ❤❛s ❞✐♠❡♥s✐♦♥ ♦♥❡✮❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ❝❧❛ss✐❝❛❧ s❡q✉❡♥❝❡ s♣❛❝❡s ℓ ♣ ( ✶ � ♣ < ∞ ) ❛♥❞ ❝ ✵ ; ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡s ▲ ♣ [ ✵ , ✶ ] ( ✶ < ♣ < ∞ ); ◮ t❤❡ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ❞ ✇ , ♣ ❛♥❞ t❤❡ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ❤ ▼ ; ◮ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ ❊ ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❡❞ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ❡①❛♠♣❧❡s✿ ◮ t❤❡ ▲❡❜❡s❣✉❡ s♣❛❝❡ ▲ ✶ [ ✵ , ✶ ]; ◮ t❤❡ s♣❛❝❡ ❈ ( ❑ ) ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❛♥ ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ❑ ❀ ◮ t❤❡ ♣ t❤ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡ ❏ ♣ ( ✶ < ♣ < ∞ ) ✳ ❍❡♥❝❡ ♥♦t ❛❧❧ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❛r❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝❛s❡s✳ P❡rs♣❡❝t✐✈❡✳ ●♦✇❡rs✬s ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠✿ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ✐s ❡✐t❤❡r ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ♥♦♥❡ ♦❢ ✐ts ❝❧♦s❡❞ s✉❜s♣❛❝❡s ❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ t❤❡ ❞✐r❡❝t s✉♠ ♦❢ t✇♦ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲ ❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✮✱ ♦r ✐t ❝♦♥t❛✐♥s ❛ s✉❜s♣❛❝❡ ✇❤✐❝❤ ❤❛s ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❙❝❤❛✉❞❡r ❜❛s✐s✳ ✺
✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ ①✳ ▼♦r❡♦✈❡r✱ ✵ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ① ② ❊ ① ② ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ① ❢♦r s♦♠❡ ① ✵ ✐s ❊ ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❦❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❝♦rr❡s♣♦♥❞s t♦ ❊ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✻
▼♦r❡♦✈❡r✱ ✵ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ① ② ❊ ① ② ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ① ❢♦r s♦♠❡ ① ✵ ✐s ❊ ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❦❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❝♦rr❡s♣♦♥❞s t♦ ❊ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ✻
❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ① ❢♦r s♦♠❡ ① ✵ ✐s ❊ ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❦❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❝♦rr❡s♣♦♥❞s t♦ ❊ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ▼♦r❡♦✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ✻
◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ✜①❡❞❄ ❊ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❦❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❝♦rr❡s♣♦♥❞s t♦ ❊ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ▼♦r❡♦✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ML ① ❢♦r s♦♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ✻
❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❦❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❝♦rr❡s♣♦♥❞s t♦ ❊ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ▼♦r❡♦✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ML ① ❢♦r s♦♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✜①❡❞❄ ✻
Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❦❡r ❚ ❋ ❚ ❊ ❋ ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❝♦rr❡s♣♦♥❞s t♦ ❊ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ▼♦r❡♦✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ML ① ❢♦r s♦♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ✻
❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❝♦rr❡s♣♦♥❞s t♦ ❊ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ▼♦r❡♦✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ML ① ❢♦r s♦♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❋ �→ { ❚ ∈ B ( ❊ ) : ❋ ⊆ ❦❡r ❚ } ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ✳ ✻
❛♥❞ t❤❡r❡❢♦r❡ ✐s ✜①❡❞✳ ❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ▼♦r❡♦✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ML ① ❢♦r s♦♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❋ �→ { ❚ ∈ B ( ❊ ) : ❋ ⊆ ❦❡r ❚ } ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ❝♦rr❡s♣♦♥❞s t♦ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ✻
❆ r❡✜♥❡♠❡♥t ♦❢ t❤❡ q✉❡st✐♦♥ ❖❜s❡r✈❛t✐♦♥✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡✳ ❋♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✱ ML ① = { ❚ ∈ B ( ❊ ) : ❚① = ✵ } ✐s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✐♥❣❧❡ ♣r♦❥❡❝t✐♦♥ ■ − P✱ ✇❤❡r❡ P ✐s ❛♥② ♣r♦❥❡❝t✐♦♥ ♦❢ ❊ ♦♥t♦ C ①✳ ▼♦r❡♦✈❡r✱ ( ① , ② ∈ ❊ \ { ✵ } ) . ML ① = ML ② ⇐ ⇒ ① ❛♥❞ ② ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ ❚❡r♠✐♥♦❧♦❣②✳ ❆ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ t❤❡ ❢♦r♠ ML ① ❢♦r s♦♠❡ ① ∈ ❊ \ { ✵ } ✐s ✜①❡❞ ✳ ◗✉❡st✐♦♥ ■■✳ ■s ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✜①❡❞❄ ❋❛❝t✳ ❚❤✐s ✐s tr✉❡ ❢♦r ❊ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ Pr♦♦❢✳ ❚❤❡ ♠❛♣♣✐♥❣ ❋ �→ { ❚ ∈ B ( ❊ ) : ❋ ⊆ ❦❡r ❚ } ✐s ❛♥ ❛♥t✐✲✐s♦♠♦r♣❤✐s♠ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧✐♥❡❛r s✉❜s♣❛❝❡s ♦❢ ❊ ♦♥t♦ t❤❡ ❧❛tt✐❝❡ ♦❢ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ❝♦rr❡s♣♦♥❞s t♦ ❛ ✉♥✐q✉❡ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ✱ ❛♥❞ t❤❡r❡❢♦r❡ L ✐s ✜①❡❞✳ ✷ ✻
❍❡♥❝❡ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❊ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✳ ✵ ✳ ❋❛❝t✳ ❊ ① ❢♦r ❡❛❝❤ ① ❊ Pr♦♦❢✳ ●✐✈❡♥ ① ✵ ✱ t❛❦❡ s✉❝❤ t❤❛t ① ✶✳ ❚❤❡♥ ❊ ❊ ❚ ② ② ① ❊ ❊ ❜❡❧♦♥❣s t♦ ❊ ✱ ❜✉t ♥♦t t♦ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ✼
✵ ✳ ❋❛❝t✳ ❊ ① ❢♦r ❡❛❝❤ ① ❊ Pr♦♦❢✳ ●✐✈❡♥ ① ✵ ✱ t❛❦❡ s✉❝❤ t❤❛t ① ✶✳ ❚❤❡♥ ❊ ❊ ❚ ② ② ① ❊ ❊ ❜❡❧♦♥❣s t♦ ❊ ✱ ❜✉t ♥♦t t♦ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ✼
Pr♦♦❢✳ ●✐✈❡♥ ① ✵ ✱ t❛❦❡ s✉❝❤ t❤❛t ① ✶✳ ❚❤❡♥ ❊ ❊ ❚ ② ② ① ❊ ❊ ❜❡❧♦♥❣s t♦ ❊ ✱ ❜✉t ♥♦t t♦ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) �⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✳ ✼
❚❤❡♥ ❚ ② ② ① ❊ ❊ ❜❡❧♦♥❣s t♦ ❊ ✱ ❜✉t ♥♦t t♦ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) �⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t � ① , λ � = ✶✳ ✼
❜✉t ♥♦t t♦ ① ❜❡❝❛✉s❡ ❚① ✵✳ ① ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) �⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t � ① , λ � = ✶✳ ❚❤❡♥ ❚ : ② �→ � ② , λ � ① , ❊ → ❊ , ❜❡❧♦♥❣s t♦ F ( ❊ ) ✱ ✼
❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) �⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t � ① , λ � = ✶✳ ❚❤❡♥ ❚ : ② �→ � ② , λ � ① , ❊ → ❊ , ❜❡❧♦♥❣s t♦ F ( ❊ ) ✱ ❜✉t ♥♦t t♦ ML ① ❜❡❝❛✉s❡ ❚① = ① � = ✵✳ ✷ ✼
▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞✳ ❚❤❡♥ ❊ ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) �⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t � ① , λ � = ✶✳ ❚❤❡♥ ❚ : ② �→ � ② , λ � ① , ❊ → ❊ , ❜❡❧♦♥❣s t♦ F ( ❊ ) ✱ ❜✉t ♥♦t t♦ ML ① ❜❡❝❛✉s❡ ❚① = ① � = ✵✳ ✷ ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ✼
◗✉❡st✐♦♥ ■■■✳ ■s ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ❊ ♦❢ ❊ ❄ ◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) �⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t � ① , λ � = ✶✳ ❚❤❡♥ ❚ : ② �→ � ② , λ � ① , ❊ → ❊ , ❜❡❧♦♥❣s t♦ F ( ❊ ) ✱ ❜✉t ♥♦t t♦ ML ① ❜❡❝❛✉s❡ ❚① = ① � = ✵✳ ✷ ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ❚❤❡♥ B ( ❊ ) ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ✼
◗✉❡st✐♦♥ ■■ ✖ t❤❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥ F ( ❊ ) = { ❚ ∈ B ( ❊ ) : ❞✐♠ ❚ ( ❊ ) < ∞} ✐s ❛ ♣r♦♣❡r✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ F ( ❊ ) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❋❛❝t✳ F ( ❊ ) �⊆ ML ① ❢♦r ❡❛❝❤ ① ∈ ❊ \ { ✵ } ✳ Pr♦♦❢✳ ●✐✈❡♥ ① ∈ ❊ \ { ✵ } ✱ t❛❦❡ λ ∈ ❊ ∗ s✉❝❤ t❤❛t � ① , λ � = ✶✳ ❚❤❡♥ ❚ : ② �→ � ② , λ � ① , ❊ → ❊ , ❜❡❧♦♥❣s t♦ F ( ❊ ) ✱ ❜✉t ♥♦t t♦ ML ① ❜❡❝❛✉s❡ ❚① = ① � = ✵✳ ✷ ❈♦r♦❧❧❛r②✳ ❆ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■ ✐♠♣❧✐❡s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r t♦ ◗✳ ■✿ ▲❡t ❊ ❜❡ ❛♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❇❛♥❛❝❤ s♣❛❝❡✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t ❡✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ❚❤❡♥ B ( ❊ ) ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ✇❤✐❝❤ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✳ ◗✉❡st✐♦♥ ■■■✳ ■s F ( ❊ ) ❡✈❡r ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ❄ ✼
✭✐✐✮ ❝♦♥t❛✐♥s ❊ ✳ ❚❤❡ s❡t ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ❊ ♦❢ ❊ ✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♦r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜❡❧♦♥❣s t♦ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧ ♦❢ t❤❡ ❈❛❧❦✐♥ ❛❧❣❡❜r❛ ❊ ✳ ❚ ❊ ❊ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♦r ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ❞✐♠ ❦❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♦r ❡❛❝❤ ❙ ∈ B ( ❊ ) ✳ ✽
✭✐✐✮ ❝♦♥t❛✐♥s ❊ ✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♦r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❜❡❧♦♥❣s t♦ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧ ♦❢ t❤❡ ❈❛❧❦✐♥ ❛❧❣❡❜r❛ ❊ ✳ ❚ ❊ ❊ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♦r ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ❞✐♠ ❦❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♦r ❡❛❝❤ ❙ ∈ B ( ❊ ) ✳ ❚❤❡ s❡t E ( ❊ ) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ✽
✭✐✐✮ ❝♦♥t❛✐♥s ❊ ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♦r ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ❞✐♠ ❦❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♦r ❡❛❝❤ ❙ ∈ B ( ❊ ) ✳ ❚❤❡ s❡t E ( ❊ ) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♦r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♦♥❣s t♦ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧ ♦❢ t❤❡ ❈❛❧❦✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ✽
✭✐✐✮ ❝♦♥t❛✐♥s ❊ ✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ ✐s ✜①❡❞❀ ♦r ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ❞✐♠ ❦❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♦r ❡❛❝❤ ❙ ∈ B ( ❊ ) ✳ ❚❤❡ s❡t E ( ❊ ) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♦r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♦♥❣s t♦ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧ ♦❢ t❤❡ ❈❛❧❦✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ✽
✭✐✐✮ ❝♦♥t❛✐♥s ❊ ✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ❞✐♠ ❦❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♦r ❡❛❝❤ ❙ ∈ B ( ❊ ) ✳ ❚❤❡ s❡t E ( ❊ ) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♦r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♦♥❣s t♦ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧ ♦❢ t❤❡ ❈❛❧❦✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✽
❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❚ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❊ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ■ − ❙❚ ✐s ❛ ❋r❡❞❤♦❧♠ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ❞✐♠ ❦❡r ( ■ − ❙❚ ) < ∞ ❛♥❞ ❞✐♠ ( ■ − ❙❚ )( ❊ ) < ∞ , ❢♦r ❡❛❝❤ ❙ ∈ B ( ❊ ) ✳ ❚❤❡ s❡t E ( ❊ ) ♦❢ ✐♥❡ss❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦♥ ❊ ✐s ❛ ❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✱ ❛♥❞ ♣r♦♣❡r ✇❤❡♥❡✈❡r ❊ ✐s ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ❆♥ ❡q✉✐✈❛❧❡♥t✱ ♠♦r❡ ❛❧❣❡❜r❛✐❝✱ ❞❡✜♥✐t✐♦♥ ✐s t❤❛t ❚ ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❚ + K ( ❊ ) ❜❡❧♦♥❣s t♦ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧ ♦❢ t❤❡ ❈❛❧❦✐♥ ❛❧❣❡❜r❛ B ( ❊ ) / K ( ❊ ) ✳ ❊①❛♠♣❧❡✳ ❆ ❍✐❧❜❡rt✲s♣❛❝❡ ♦♣❡r❛t♦r ✐s ✐♥❡ss❡♥t✐❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E ( ❊ ) ✳ ✽
■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼ ✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ▼ ✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼ ✳ ❘❡♠❛r❦s✳ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦♥t❛✐♥s ❊ ✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E ( ❊ ) ✳ ✾
■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼ ✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ▼ ✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼ ✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦♥t❛✐♥s ❊ ✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❦s✳ ◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳ ✾
❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦♥t❛✐♥s ❊ ✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❦s✳ ◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳ ◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼ ✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ ▼ ✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼ ✳ ✾
❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ❝♦♥t❛✐♥s ❊ ✳ ❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❦s✳ ◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳ ◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼ ✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ ▼ ✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼ ✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ✾
❚❤❡ ❡q✉✐✈❛❧❡♥❝❡ ♦❢ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❘❡❝❛❧❧✿ ▲❡t ❊ ❜❡ ❛ ♥♦♥✲③❡r♦ ❇❛♥❛❝❤ s♣❛❝❡✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ L ♦❢ B ( ❊ ) ✱ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿ ✭✐✮ L ✐s ✜①❡❞❀ ♦r ✭✐✐✮ L ❝♦♥t❛✐♥s E ( ❊ ) ✳ ❘❡♠❛r❦s✳ ◮ ❚❤✐s r❡s✉❧t ✐s ❡ss❡♥t✐❛❧❧② ❦♥♦✇♥ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ✈❡r② ❞✐✛❡r❡♥t ❢r♦♠ ♦✉rs✳ ◮ ■t ❝❛♥ ❜❡ ✈✐❡✇❡❞ ❛s t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ ❢❛❝t t❤❛t ❛♥ ✉❧tr❛✜❧t❡r ♦♥ ❛ s❡t ▼ ✐s ❡✐t❤❡r ✜①❡❞ ✭✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ❝♦♥s✐sts ♦❢ ♣r❡❝✐s❡❧② t❤♦s❡ s✉❜s❡ts ♦❢ ▼ ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ✜①❡❞ ❡❧❡♠❡♥t ① ∈ ▼ ✮✱ ♦r ✐t ❝♦♥t❛✐♥s t❤❡ ❋ré❝❤❡t ✜❧t❡r ♦❢ ❛❧❧ ❝♦✜♥✐t❡ s✉❜s❡ts ♦❢ ▼ ✳ ❈♦r♦❧❧❛r②✳ ◗✉❡st✐♦♥s ■■ ❛♥❞ ■■■ ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥s❡✿ ❊✈❡r② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♥♦ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ❝♦♥t❛✐♥s F ( ❊ ) ✳ ✾
✭ ❡①❛♠♣❧❡s✿ ♣ ❛♥❞ ▲ ♣ ✵ ✶ ❢♦r ✶ ✮❀ ♣ ✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ✱ ❊ ❍✱ ♦r ❊ ❝ ✵ ❍✱ ✇❤❡r❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ✱ ❊ ❍✱ ♦r ❊ ❝ ✵ ❍✱ ✇❤❡r❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭ ❡①❛♠♣❧❡s✿ ♣ ❛♥❞ ▲ ♣ ✵ ✶ ❢♦r ✶ ✮❀ ♣ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ✱ ❊ ❍✱ ♦r ❊ ❝ ✵ ❍✱ ✇❤❡r❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✱ ❊ ❍✱ ♦r ❊ ❝ ✵ ❍✱ ✇❤❡r❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭ ❡①❛♠♣❧❡s✿ ✮❀ ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✱ ❊ ❍✱ ♦r ❊ ❝ ✵ ❍✱ ✇❤❡r❡ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✐✐✐✮ ❊ ❝ ✵ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✱ ❊ ❍✱ ♦r ❊ ❝ ✵ ❍ ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✱ ♦r ❊ ❝ ✵ ❍ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍ ✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♦r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t ✳ ❙① P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♦r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ✶✵
✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♦r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ε > ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t � ❙① � � ε ✳ ✶✵
✭✈✮ ❊ ❈ ❑ ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ❑ ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♦r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ε > ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t � ❙① � � ε ✳ ✶✵
P♦s✐t✐✈❡ ❛♥s✇❡rs t♦ ◗✉❡st✐♦♥ ■■ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ▲❡t ❊ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ✇❤✐❝❤ s❛t✐s✜❡s ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜✈❡ ❝♦♥❞✐t✐♦♥s✿ ✭✐✮ ❊ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s ❛♥❞ ✐s ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ✐ts ❜✐❞✉❛❧ ✭ ❡①❛♠♣❧❡s✿ ℓ ♣ ❛♥❞ ▲ ♣ [ ✵ , ✶ ] ❢♦r ✶ � ♣ < ∞ ✮❀ ✭✐✐✮ ❊ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❊ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♠♣❧❡♠❡♥t❡❞ ✐♥ ❛♥② s✉♣❡rs♣❛❝❡ ✭ ❡①❛♠♣❧❡s✿ ℓ ∞ (Γ) ❢♦r s♦♠❡ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t Γ ✮❀ ✭✐✐✐✮ ❊ = ❝ ✵ (Γ) ✱ ❊ = ❍✱ ♦r ❊ = ❝ ✵ (Γ) ⊕ ❍✱ ✇❤❡r❡ Γ ✐s ❛ ♥♦♥✲❡♠♣t② ✐♥❞❡① s❡t ❛♥❞ ❍ ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡❀ ✭✐✈✮ ❊ ❤❛s ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❊ ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t② ✭❡①❛♠♣❧❡s✿ ❤❡r❡❞✐t❛r✐❧② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡s✮❀ ✭✈✮ ❊ = ❈ ( ❑ ) ✱ ✇❤❡r❡ ❑ ✐s ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ✇✐t❤♦✉t ✐s♦❧❛t❡❞ ♣♦✐♥ts✱ ❛♥❞ ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❈ ( ❑ ) ✐s ❛ ✇❡❛❦ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ✐s ❛ str✐❝t❧② s✐♥❣✉❧❛r ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦r✳ ❚❤❡♥ ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✐s ✜①❡❞✳ ❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ❙ ♦♥ ❊ ✐s str✐❝t❧② s✐♥❣✉❧❛r ✐❢✱ ❢♦r ❡❛❝❤ ε > ✵✱ ❡❛❝❤ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ♦❢ ❊ ❝♦♥t❛✐♥s ❛ ✉♥✐t ✈❡❝t♦r ① s✉❝❤ t❤❛t � ❙① � � ε ✳ ✶✵
❆ st❛t❡ ♦♥ ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ ♦♥ s✉❝❤ t❤❛t ✵ ❛ ❛ ❛ ●✐✈❡♥ ❛ st❛t❡ ♦♥ ✱ t❤❡ s❡t ✵ ❛ ❛ ❛ ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ ✳ ❙✉♣♣♦s❡ t❤❛t ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♦r✳ ❚❤❡♥ ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍ ✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ �→ ❛ ⋆ ✳ ✶✶
●✐✈❡♥ ❛ st❛t❡ ♦♥ ✱ t❤❡ s❡t ✵ ❛ ❛ ❛ ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ ✳ ❙✉♣♣♦s❡ t❤❛t ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♦r✳ ❚❤❡♥ ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍ ✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ �→ ❛ ⋆ ✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t � ❛ ⋆ ❛ , λ � � ✵ ( ❛ ∈ A ) . ✶✶
■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ ✳ ❙✉♣♣♦s❡ t❤❛t ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♦r✳ ❚❤❡♥ ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍ ✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ �→ ❛ ⋆ ✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t � ❛ ⋆ ❛ , λ � � ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t N λ = { ❛ ∈ A : � ❛ ⋆ ❛ , λ � = ✵ } ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ✶✶
❙✉♣♣♦s❡ t❤❛t ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ❍ ❜❡ ❛ ✉♥✐t ❍ ✈❡❝t♦r✳ ❚❤❡♥ ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❤❛t ✐s✱ ❚ ❚① ① ❚ ❍ ✇❤❡r❡ ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍ ✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ �→ ❛ ⋆ ✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t � ❛ ⋆ ❛ , λ � � ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t N λ = { ❛ ∈ A : � ❛ ⋆ ❛ , λ � = ✵ } ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ∗ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ A ✳ ✶✶
❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ ♦♥ ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r ❦❡r ✱ ✐♥ ✇❤✐❝❤ ❍ ❍ ❝❛s❡ ✳ ❍ ❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ �→ ❛ ⋆ ✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t � ❛ ⋆ ❛ , λ � � ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t N λ = { ❛ ∈ A : � ❛ ⋆ ❛ , λ � = ✵ } ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ∗ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ A ✳ ❙✉♣♣♦s❡ t❤❛t A = B ( ❍ ) ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ∈ ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥ N λ = ML ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❤❛t ✐s✱ � ❚ , λ � = ( ❚① | ① ) ( ❚ ∈ B ( ❍ )) , ✇❤❡r❡ ( · | · ) ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍ ✳ ✶✶
❚❤❡ ❍✐❧❜❡rt✲s♣❛❝❡ ❝❛s❡ ▲❡t A ❜❡ ❛ ✉♥✐t❛❧ ❈ ∗ ✲❛❧❣❡❜r❛✱ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❛ �→ ❛ ⋆ ✳ ❆ st❛t❡ ♦♥ A ✐s ❛ ♥♦r♠✲♦♥❡ ❢✉♥❝t✐♦♥❛❧ λ ♦♥ A s✉❝❤ t❤❛t � ❛ ⋆ ❛ , λ � � ✵ ( ❛ ∈ A ) . ●✐✈❡♥ ❛ st❛t❡ λ ♦♥ A ✱ t❤❡ s❡t N λ = { ❛ ∈ A : � ❛ ⋆ ❛ , λ � = ✵ } ✐s ❛ ❝❧♦s❡❞ ❧❡❢t ✐❞❡❛❧ ♦❢ A ✳ ■t ✐s ♠❛①✐♠❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s ❛ ♣✉r❡ st❛t❡ ✱ t❤❛t ✐s✱ ❛♥ ❡①tr❡♠❡ ♣♦✐♥t ♦❢ t❤❡ ✇❡❛❦ ∗ ✲❝♦♠♣❛❝t✱ ❝♦♥✈❡① s❡t ♦❢ ❛❧❧ st❛t❡s ♦♥ A ✳ ❙✉♣♣♦s❡ t❤❛t A = B ( ❍ ) ❢♦r s♦♠❡ ❍✐❧❜❡rt s♣❛❝❡ ❍ ✱ ❛♥❞ ❧❡t ① ∈ ❍ ❜❡ ❛ ✉♥✐t ✈❡❝t♦r✳ ❚❤❡♥ N λ = ML ① ✐❢ ❛♥❞ ♦♥❧② ✐❢ λ ✐s t❤❡ ✈❡❝t♦r st❛t❡ ✐♥❞✉❝❡❞ ❜② ① ✱ t❤❛t ✐s✱ � ❚ , λ � = ( ❚① | ① ) ( ❚ ∈ B ( ❍ )) , ✇❤❡r❡ ( · | · ) ❞❡♥♦t❡s t❤❡ ✐♥♥❡r ♣r♦❞✉❝t ♦♥ ❍ ✳ ❚❤❡ ❉✐❝❤♦t♦♠② ❚❤❡♦r❡♠ ❢♦r ❍✐❧❜❡rt s♣❛❝❡s ❢♦❧❧♦✇s ❢r♦♠ t❤❡s❡ ❢❛❝ts ❜❡❝❛✉s❡ ❡❛❝❤ ♣✉r❡ st❛t❡ λ ♦♥ B ( ❍ ) ✐s ❡✐t❤❡r ❛ ✈❡❝t♦r st❛t❡✱ ♦r K ( ❍ ) ⊆ ❦❡r λ ✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ K ( ❍ ) ⊆ N λ ✳ ✶✶
✭✐✐✮ ❳ ❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳ ❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✶ ✳ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ ❆r❣②r♦s✕❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳ ❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳ ❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✶✷
✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳ ❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✶ ✳ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ ❆r❣②r♦s✕❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳ ❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳ ❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳ ❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✶✷
❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ ❆r❣②r♦s✕❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❚❤❡♦r❡♠ ✭❆r❣②r♦s✕❍❛②❞♦♥ ✷✵✶✶✮✳ ❚❤❡r❡ ✐s ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ✇❤✐❝❤ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ♣r♦♣❡rt✐❡s✿ ✭✐✮ ❳ ❆❍ ❤❛s ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❡❛❝❤ ♦♣❡r❛t♦r ♦♥ ❳ ❆❍ ✐s ❛ ❝♦♠♣❛❝t ♣❡rt✉r❜❛t✐♦♥ ♦❢ ❛ s❝❛❧❛r ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ✐❞❡♥t✐t②❀ ✭✐✐✮ ❳ ❆❍ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❳ ❆❍ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ℓ ✶ ✳ ✶✷
❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ✷ ✷ ✲♠❛tr✐❝❡s ❚ ✶ ✶ ❳ ❆❍ ❳ ❆❍ ❚ ✶ ✷ ❳ ❆❍ ❚ ✷ ✶ ❳ ❆❍ ❚ ✷ ✷ ❑❡② ♣♦✐♥t✿ ❚ ✶ ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✶ ✶ ✐s ❝♦♠♣❛❝t ❊ ✶ ❚ ✷ ✶ ❚ ✷ ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ ❊ ✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ✵ ✵ ▲ ❱❯ ❲ ✇❤❡r❡ ❳ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ✶✸
❑❡② ♣♦✐♥t✿ ❚ ✶ ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✶ ✶ ✐s ❝♦♠♣❛❝t ❊ ✶ ❚ ✷ ✶ ❚ ✷ ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ ❊ ✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ✵ ✵ ▲ ❱❯ ❲ ✇❤❡r❡ ❳ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s � ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ � ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ✶✸
❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✶ ✶ ✐s ❝♦♠♣❛❝t ❊ ✶ ❚ ✷ ✶ ❚ ✷ ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ ❊ ✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ✵ ✵ ▲ ❱❯ ❲ ✇❤❡r❡ ❳ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s � ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ � ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♦✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ✶✸
✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ✵ ✵ ▲ ❱❯ ❲ ✇❤❡r❡ ❳ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s � ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ � ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♦✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t �� ❚ ✶ , ✶ � � ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♦♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B ( ❊ ) ✶✸
■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ✵ ✵ ▲ ❱❯ ❲ ✇❤❡r❡ ❳ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s � ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ � ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♦✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t �� ❚ ✶ , ✶ � � ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♦♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ✶✸
✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ✵ ✵ ▲ ❱❯ ❲ ✇❤❡r❡ ❳ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s � ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ � ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♦✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t �� ❚ ✶ , ✶ � � ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♦♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞ ✶✸
▼♦r❡ ♣r❡❝✐s❡❧②✱ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r ✵ ✵ ▲ ❱❯ ❲ ✇❤❡r❡ ❳ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ ❳ ❆❍ ✱ ❳ ❆❍ ✶ ✷ ✶ ❛♥❞ ❲ ✷ ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ❱ ✶ ❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s � ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ � ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♦✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t �� ❚ ✶ , ✶ � � ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♦♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ✶✸
❆ ♥❡❣❛t✐✈❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■■✿ t❤❡ ❡①❛♠♣❧❡ ▲❡t ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❲❡ ✐❞❡♥t✐❢② ♦♣❡r❛t♦rs ❚ ♦♥ ❊ ✇✐t❤ ( ✷ × ✷ ) ✲♠❛tr✐❝❡s � ❚ ✶ , ✶ : ❳ ❆❍ → ❳ ❆❍ � ❚ ✶ , ✷ : ℓ ∞ → ❳ ❆❍ . ❚ ✷ , ✶ : ❳ ❆❍ → ℓ ∞ ❚ ✷ , ✷ : ℓ ∞ → ℓ ∞ ❑❡② ♣♦✐♥t✿ ❚ ✶ , ✷ ✐s ♥❡❝❡ss❛r✐❧② str✐❝t❧② s✐♥❣✉❧❛r✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ s❡t �� ❚ ✶ , ✶ � � ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✶ , ✶ ✐s ❝♦♠♣❛❝t K ✶ = ❚ ✷ , ✶ ❚ ✷ , ✷ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ B ( ❊ ) ✱ ❛♥❞ ❤❡♥❝❡ ❛❧s♦ ❛ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧✳ ■t ✐s ♥♦t ✜①❡❞✱ ❜✉t ✐t ✐s s✐♥❣❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ K ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ ♦♣❡r❛t♦r � ✵ ✵ � ▲ = , ❱❯ ∗ κ ❲ ✇❤❡r❡ κ : ❳ ❆❍ → ❳ ∗∗ ❆❍ ✐s t❤❡ ❝❛♥♦♥✐❝❛❧ ❡♠❜❡❞❞✐♥❣✱ ✇❤✐❧❡ ❯ : ℓ ✶ → ❳ ∗ ❆❍ ✱ ❱ : ℓ ∗ ✶ = ℓ ∞ → ℓ ∞ ( ✷ N − ✶ ) ❛♥❞ ❲ : ℓ ∞ → ℓ ∞ ( ✷ N ) ❛r❡ ✐s♦♠♦r♣❤✐s♠s✳ ✶✸
❍❡♥❝❡ ❚ ✶ ✶ ❚ ✶ ✷ ❚ ✷ ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r ❊ ❚ ✷ ✶ ❚ ✷ ✷ ✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ ❊ ✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ ❊ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳ ❍♦✇ ❛❜♦✉t ◗✉❡st✐♦♥ ■❄ ❘❡❝❛❧❧✿ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K ✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ✶✹
■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳ ❍♦✇ ❛❜♦✉t ◗✉❡st✐♦♥ ■❄ ❘❡❝❛❧❧✿ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K ✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ �� ❚ ✶ , ✶ � � ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✷ , ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r , ❚ ✷ , ✶ ❚ ✷ , ✷ ✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ✶✹
❍♦✇ ❛❜♦✉t ◗✉❡st✐♦♥ ■❄ ❘❡❝❛❧❧✿ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳ ❚❤❡♦r❡♠ ✭❉❑❑❑▲✮✳ ❚❤❡ ✐❞❡❛❧ K ✶ ✐s t❤❡ ✉♥✐q✉❡ ♥♦♥✲✜①❡❞✱ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ❍❡♥❝❡ �� ❚ ✶ , ✶ � � ❚ ✶ , ✷ ∈ B ( ❊ ) : ❚ ✷ , ✷ ✐s str✐❝t❧② s✐♥❣✉❧❛r , ❚ ✷ , ✶ ❚ ✷ , ✷ ✇❤✐❝❤ ✐s ❛ ♠❛①✐♠❛❧ t✇♦✲s✐❞❡❞ ✐❞❡❛❧ ♦❢ B ( ❊ ) ✱ ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❛♥s✇❡r t♦ ◗✉❡st✐♦♥ ■ ✐s ♣♦s✐t✐✈❡ ❢♦r ❊✳ ✶✹
✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❛❜♦✈❡✱ ❊ ✳✮ ❳ ❆❍ ❆♥s✇❡r✿ ②❡s✦ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♦s ❛♥❞ ❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ❝♦♥t❛✐♥s ❛ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ❨ ♦❢ ✐♥✜♥✐t❡ ❝♦❞✐♠❡♥s✐♦♥ s✉❝❤ t❤❛t✿ ✭✐✮ ❡❛❝❤ ♦♣❡r❛t♦r ❢r♦♠ ❨ ✐♥t♦ ❳ ❆❍ ❤❛s t❤❡ ❢♦r♠ ❏ ❑ ❢♦r s♦♠❡ ❛♥❞ s♦♠❡ ❝♦♠♣❛❝t ♦♣❡r❛t♦r ❑✱ ✇❤❡r❡ ❏ ❨ ❳ ❆❍ ❞❡♥♦t❡s t❤❡ ✐♥❝❧✉s✐♦♥❀ ✭✐✐✮ ❨ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❨ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B ( ❊ ) ❝♦♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✶✺
❆♥s✇❡r✿ ②❡s✦ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♦s ❛♥❞ ❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ❝♦♥t❛✐♥s ❛ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ❨ ♦❢ ✐♥✜♥✐t❡ ❝♦❞✐♠❡♥s✐♦♥ s✉❝❤ t❤❛t✿ ✭✐✮ ❡❛❝❤ ♦♣❡r❛t♦r ❢r♦♠ ❨ ✐♥t♦ ❳ ❆❍ ❤❛s t❤❡ ❢♦r♠ ❏ ❑ ❢♦r s♦♠❡ ❛♥❞ s♦♠❡ ❝♦♠♣❛❝t ♦♣❡r❛t♦r ❑✱ ✇❤❡r❡ ❏ ❨ ❳ ❆❍ ❞❡♥♦t❡s t❤❡ ✐♥❝❧✉s✐♦♥❀ ✭✐✐✮ ❨ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❨ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B ( ❊ ) ❝♦♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❛❜♦✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ✶✺
❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♦s ❛♥❞ ❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ❝♦♥t❛✐♥s ❛ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ❨ ♦❢ ✐♥✜♥✐t❡ ❝♦❞✐♠❡♥s✐♦♥ s✉❝❤ t❤❛t✿ ✭✐✮ ❡❛❝❤ ♦♣❡r❛t♦r ❢r♦♠ ❨ ✐♥t♦ ❳ ❆❍ ❤❛s t❤❡ ❢♦r♠ ❏ ❑ ❢♦r s♦♠❡ ❛♥❞ s♦♠❡ ❝♦♠♣❛❝t ♦♣❡r❛t♦r ❑✱ ✇❤❡r❡ ❏ ❨ ❳ ❆❍ ❞❡♥♦t❡s t❤❡ ✐♥❝❧✉s✐♦♥❀ ✭✐✐✮ ❨ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❨ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B ( ❊ ) ❝♦♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❛❜♦✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✦ ✶✺
✭✐✐✮ ❨ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❨ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B ( ❊ ) ❝♦♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❛❜♦✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✦ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♦s ❛♥❞ ❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ❝♦♥t❛✐♥s ❛ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ❨ ♦❢ ✐♥✜♥✐t❡ ❝♦❞✐♠❡♥s✐♦♥ s✉❝❤ t❤❛t✿ ✭✐✮ ❡❛❝❤ ♦♣❡r❛t♦r ❢r♦♠ ❨ ✐♥t♦ ❳ ❆❍ ❤❛s t❤❡ ❢♦r♠ α ❏ + ❑ ❢♦r s♦♠❡ α ∈ C ❛♥❞ s♦♠❡ ❝♦♠♣❛❝t ♦♣❡r❛t♦r ❑✱ ✇❤❡r❡ ❏ : ❨ → ❳ ❆❍ ❞❡♥♦t❡s t❤❡ ✐♥❝❧✉s✐♦♥❀ ✶✺
✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❨ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ✶ ✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B ( ❊ ) ❝♦♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❛❜♦✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✦ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♦s ❛♥❞ ❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ❝♦♥t❛✐♥s ❛ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ❨ ♦❢ ✐♥✜♥✐t❡ ❝♦❞✐♠❡♥s✐♦♥ s✉❝❤ t❤❛t✿ ✭✐✮ ❡❛❝❤ ♦♣❡r❛t♦r ❢r♦♠ ❨ ✐♥t♦ ❳ ❆❍ ❤❛s t❤❡ ❢♦r♠ α ❏ + ❑ ❢♦r s♦♠❡ α ∈ C ❛♥❞ s♦♠❡ ❝♦♠♣❛❝t ♦♣❡r❛t♦r ❑✱ ✇❤❡r❡ ❏ : ❨ → ❳ ❆❍ ❞❡♥♦t❡s t❤❡ ✐♥❝❧✉s✐♦♥❀ ✭✐✐✮ ❨ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✶✺
❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡ ❊ s✉❝❤ t❤❛t B ( ❊ ) ❝♦♥t❛✐♥s ❛ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧❄ ✭❘❡❝❛❧❧✿ ✐♥ t❤❡ ❡①❛♠♣❧❡✱ ❛❜♦✈❡✱ ❊ = ❳ ❆❍ ⊕ ℓ ∞ ✳✮ ❆♥s✇❡r✿ ②❡s✦ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ❆r❣②r♦s ❛♥❞ ❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❆❍ ❝♦♥t❛✐♥s ❛ ❝❧♦s❡❞✱ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡ ❨ ♦❢ ✐♥✜♥✐t❡ ❝♦❞✐♠❡♥s✐♦♥ s✉❝❤ t❤❛t✿ ✭✐✮ ❡❛❝❤ ♦♣❡r❛t♦r ❢r♦♠ ❨ ✐♥t♦ ❳ ❆❍ ❤❛s t❤❡ ❢♦r♠ α ❏ + ❑ ❢♦r s♦♠❡ α ∈ C ❛♥❞ s♦♠❡ ❝♦♠♣❛❝t ♦♣❡r❛t♦r ❑✱ ✇❤❡r❡ ❏ : ❨ → ❳ ❆❍ ❞❡♥♦t❡s t❤❡ ✐♥❝❧✉s✐♦♥❀ ✭✐✐✮ ❨ ❤❛s ❛ ❙❝❤❛✉❞❡r ❜❛s✐s❀ ✭✐✐✐✮ t❤❡ ❞✉❛❧ s♣❛❝❡ ♦❢ ❨ ✐s ✐s♦♠♦r♣❤✐❝ t♦ ℓ ✶ ✳ ✶✺
❜✉t ✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ ❊ ✱ ♥❛♠❡❧② ✵ ✵ ❚ ❊ ❛♥❞ ❚ ❊ ✶ ✷ ✷ ✷ ✶ ✶ ✭✐✐✮ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ✵ ✵ ■ ❳ ❆❍ ❏ ❛♥❞ ✵ ✵ ✵ ✵ ✭✐✐✐✮ ✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B ( ❊ ) ❤❛s t❤❡ ❢♦r♠ � α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ � α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❨ + ❑ ✷ , ✷ ✇❤❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♦♠♣❛❝t✳ ✶✻
❜✉t ✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊ ✭✐✐✮ ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ✵ ✵ ■ ❳ ❆❍ ❏ ❛♥❞ ✵ ✵ ✵ ✵ ✭✐✐✐✮ ✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B ( ❊ ) ❤❛s t❤❡ ❢♦r♠ � α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ � α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❨ + ❑ ✷ , ✷ ✇❤❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✶✻
❜✉t ✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊ ✭✐✐✐✮ ✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B ( ❊ ) ❤❛s t❤❡ ❢♦r♠ � α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ � α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❨ + ❑ ✷ , ✷ ✇❤❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✭✐✐✮ M ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ✵ � ✵ � ■ ❳ ❆❍ � � ❏ ❛♥❞ , ✵ ✵ ✵ ✵ ✶✻
✭✐✐✐✮ ✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B ( ❊ ) ❤❛s t❤❡ ❢♦r♠ � α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ � α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❨ + ❑ ✷ , ✷ ✇❤❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✭✐✐✮ M ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ✵ � ✵ � ■ ❳ ❆❍ � � ❏ ❛♥❞ , ✵ ✵ ✵ ✵ ❜✉t M ✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊ ; ✶✻
❆ s❡♣❛r❛❜❧❡ ❡①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮ ▲❡t ❊ = ❳ ❆❍ ⊕ ❨ ✳ ❚❤❡♥ ❡❛❝❤ ❚ ∈ B ( ❊ ) ❤❛s t❤❡ ❢♦r♠ � α ✶ , ✶ ■ ❳ ❆❍ + ❑ ✶ , ✶ � α ✶ , ✷ ❏ + ❑ ✶ , ✷ ❚ = , ❑ ✷ , ✶ α ✷ , ✷ ■ ❨ + ❑ ✷ , ✷ ✇❤❡r❡ α ✶ , ✶ , α ✶ , ✷ , α ✷ , ✷ ∈ C ❛♥❞ t❤❡ ♦♣❡r❛t♦rs ❑ ✶ , ✶ , ❑ ✶ , ✷ , ❑ ✷ , ✶ , ❑ ✷ , ✷ ❛r❡ ❝♦♠♣❛❝t✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ✭✐✮ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♥♦♥✲✜①❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ B ( ❊ ) ✱ ♥❛♠❡❧② M ✶ = { ❚ ∈ B ( ❊ ) : α ✷ , ✷ = ✵ } M ✷ = { ❚ ∈ B ( ❊ ) : α ✶ , ✶ = ✵ } ; ❛♥❞ ✭✐✐✮ M ✶ ✐s ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② t❤❡ t✇♦ ♦♣❡r❛t♦rs ✵ � ✵ � ■ ❳ ❆❍ � � ❏ ❛♥❞ , ✵ ✵ ✵ ✵ ❜✉t M ✶ ✐s ♥♦t ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧ ❜② ❛ s✐♥❣❧❡ ♦♣❡r❛t♦r ♦♥ ❊ ; ✭✐✐✐✮ M ✷ ✐s ♥♦t ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❛s ❛ ❧❡❢t ✐❞❡❛❧✳ ✶✻
❛♥❞ r❡❢❡r❡♥❝❡s ❲❤❛t ✐s t❤❡ s✐t✉❛t✐♦♥ ❢♦r ♠❛①✐♠❛❧ r✐❣❤t ✐❞❡❛❧s ♦❢ ❊ ❄ ❑❡② r❡❢❡r❡♥❝❡s ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✽ ✭✷✵✶✸✮✱ ✷✹✺✕✷✽✻✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ■❞❡❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❖♣❡♥ ♣r♦❜❧❡♠s ◮ ▲❡t ❊ = ❈ ( ❑ ) ✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t ❈ ( ❑ ) �∼ = ❝ ✵ ✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✜①❡❞❄ ✶✼
❛♥❞ r❡❢❡r❡♥❝❡s ❑❡② r❡❢❡r❡♥❝❡s ❍✳ ●✳ ❉❛❧❡s✱ ❚✳ ❑❛♥✐❛✱ ❚✳ ❑♦❝❤❛♥❡❦✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ▼❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ♦❢ t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✽ ✭✷✵✶✸✮✱ ✷✹✺✕✷✽✻✳ ❍✳ ●✳ ❉❛❧❡s ❛♥❞ ❲✳ ➏❡❧❛③❦♦✱ ●❡♥❡r❛t♦rs ♦❢ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧s ✐♥ ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✱ ❙t✉❞✐❛ ▼❛t❤✳ ✷✶✷ ✭✷✵✶✷✮✱ ✶✼✸✕✶✾✸✳ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ■❞❡❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❖♣❡♥ ♣r♦❜❧❡♠s ◮ ▲❡t ❊ = ❈ ( ❑ ) ✱ ✇❤❡r❡ ❑ ✐s ❛♥② ✐♥✜♥✐t❡✱ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ s✉❝❤ t❤❛t ❈ ( ❑ ) �∼ = ❝ ✵ ✳ ■s ❡❛❝❤ ✜♥✐t❡❧②✲❣❡♥❡r❛t❡❞✱ ♠❛①✐♠❛❧ ❧❡❢t ✐❞❡❛❧ ♦❢ B ( ❊ ) ✜①❡❞❄ ◮ ❲❤❛t ✐s t❤❡ s✐t✉❛t✐♦♥ ❢♦r ♠❛①✐♠❛❧ r✐❣❤t ✐❞❡❛❧s ♦❢ B ( ❊ ) ❄ ✶✼
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