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Feb 27, 2024 181 likes •953 views

r tss r ts r t t r t r

SLIDE 1

▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮

❑❛③✐♠✐❡r③ ❲✐❡❧❦✐ ❯♥✐✈❡rs✐t② ✐♥ ❇②❞❣♦s③❝③

❚r❛♥s✜♥✐t❡ ♠❡t❤♦❞s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ ❛❧❣❡❜r❛s ♦❢ ♦♣❡r❛t♦rs ❇❡❞❧❡✇♦✱ ✶✽✲✷✷ ❏✉♥❡ ✷✵✶✺

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 2

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t ❘ ▲❡t ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ ✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r

✶✱ ❛♥❞ ✵

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 3

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F(X) = {f : X → ❘ : f has property F}. ▲❡t ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ ✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r

✶✱ ❛♥❞ ✵

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 4

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F(X) = {f : X → ❘ : f has property F}. ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r

✶✱ ❛♥❞ ✵

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 5

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F(X) = {f : X → ❘ : f has property F}. ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F(A) → F(X)? ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r

✶✱ ❛♥❞ ✵

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 6

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F(X) = {f : X → ❘ : f has property F}. ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F(A) → F(X)? ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r

✶✱ ❛♥❞ ✵

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 7

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F(X) = {f : X → ❘ : f has property F}. ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F(A) → F(X)? ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ F(A) s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r

✶✱ ❛♥❞ ✵

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 8

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F(X) = {f : X → ❘ : f has property F}. ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F(A) → F(X)? ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ F(A) s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r

✶✱ ❛♥❞ ✵

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 9

❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F(X) = {f : X → ❘ : f has property F}. ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F(A) → F(X)? ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ F(A) s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ Bα(A) ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s A → ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss α✱ ✇❤❡r❡ α ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r ω✶✱ ❛♥❞ B✵(A) = C(A)✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 10

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿

✐✈❡♥

♠❡tr✐③❛❜❧❡ ❛♥❞ ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 11

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿

✐✈❡♥

♠❡tr✐③❛❜❧❡ ❛♥❞ ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 12

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 13

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λ : C(A) → C(X) s✉❝❤ t❤❛t Λ r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 14

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λ : C(A) → C(X) s✉❝❤ t❤❛t Λ r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 15

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λ : C(A) → C(X) s✉❝❤ t❤❛t Λ r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 16

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λ : C(A) → C(X) s✉❝❤ t❤❛t Λ r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 17

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λ : C(A) → C(X) s✉❝❤ t❤❛t Λ r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ ❘ ♦❢ ✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ❘ ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 18

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λ : C(A) → C(X) s✉❝❤ t❤❛t Λ r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ C m(E) ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ E ⊂ ❘n ♦❢ m✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s f : ❘n → ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ❘ s✉❝❤ t❤❛t ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 19

❚❤❡ ❝❛s❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ s♣❛❝❡s ❇♦rs✉❦ ✭✶✾✸✸✮ ❛♥❞ ❉✉❣✉♥❞❥✐ ❬✸❪ ✭✶✾✺✶✮✿ ●✐✈❡♥ X ♠❡tr✐③❛❜❧❡ ❛♥❞ A ⊂ X ❝❧♦s❡❞✱ t❤❡r❡ ✐s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λ : C(A) → C(X) s✉❝❤ t❤❛t Λ r❡str✐❝t❡❞ t♦ ❜♦✉♥❞❡❞ ❢✉♥❝t✐♦♥s ✐s ❛♥ ✐s♦♠❡tr②✳ ■♥ ✶✾✼✽✱ ❆r♦♥ ❛♥❞ ❇❡r♥❡r ❬✶❪ ♦❜t❛✐♥❡❞ ❛♥❛❧♦❣♦✉s t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s✳ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s ✇❡r❡ st✉❞✐❡❞ ❜② ▼❡rr✐❡♥ ✭✶✾✻✻✮ ❬✼❪ ❛♥❞ ❇r♦♠❜❡r❣ ✭✶✾✽✷✮ ❬✷❪✳ ■♥ ✷✵✵✼✱ ❋❡✛❡r♠❛♥ ❬✹❪ ♦❜t❛✐♥❡❞ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ▼❡rr✐❡♥✬s ❛♥❞ ❇r♦♠❜❡r❣✬s r❡s✉❧ts✿ ■❢ C m(E) ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ r❡str✐❝t✐♦♥s t♦ E ⊂ ❘n ♦❢ m✲❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥s f : ❘n → ❘✱ t❤❡♥ t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ❛♥❞ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r T : C m(E) → C m(❘n) s✉❝❤ t❤❛t T(f|E) = f ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 20

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ✐s ❛ ✲s✉❜s❡t ♦❢ ✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t ♦❢

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

s✉❝❤ t❤❛t s✉♣ s✉♣ ✐♥❢ ✐♥❢ ✇❤❡♥❡✈❡r ✭✐✮ ✐s ❍❛✉s❞♦r✛ ❛♥❞ ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ ✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 21

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ✐s ❛ ✲s✉❜s❡t ♦❢ ✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t ♦❢

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

s✉❝❤ t❤❛t s✉♣ s✉♣ ✐♥❢ ✐♥❢ ✇❤❡♥❡✈❡r ✭✐✮ ✐s ❍❛✉s❞♦r✛ ❛♥❞ ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ ✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 22

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t ♦❢

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

s✉❝❤ t❤❛t s✉♣ s✉♣ ✐♥❢ ✐♥❢ ✇❤❡♥❡✈❡r ✭✐✮ ✐s ❍❛✉s❞♦r✛ ❛♥❞ ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ ✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 23

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t ♦❢

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

s✉❝❤ t❤❛t s✉♣ s✉♣ ✐♥❢ ✐♥❢ ✇❤❡♥❡✈❡r ✭✐✮ ✐s ❍❛✉s❞♦r✛ ❛♥❞ ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ ✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 24

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t ♦❢

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

s✉❝❤ t❤❛t s✉♣ s✉♣ ✐♥❢ ✐♥❢ ✇❤❡♥❡✈❡r ✭✐✮ ✐s ❍❛✉s❞♦r✛ ❛♥❞ ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ ✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 25

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t f ♦❢ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X) s✉❝❤ t❤❛t s✉♣ s✉♣ ✐♥❢ ✐♥❢ ✇❤❡♥❡✈❡r ✭✐✮ ✐s ❍❛✉s❞♦r✛ ❛♥❞ ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ ✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 26

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t f ♦❢ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X) s✉❝❤ t❤❛t s✉♣ f (A) = s✉♣ f (X) and ✐♥❢ f (A) = ✐♥❢ f (X) ✇❤❡♥❡✈❡r ✭✐✮ ✐s ❍❛✉s❞♦r✛ ❛♥❞ ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ ✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 27

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t f ♦❢ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X) s✉❝❤ t❤❛t s✉♣ f (A) = s✉♣ f (X) and ✐♥❢ f (A) = ✐♥❢ f (X) ✇❤❡♥❡✈❡r ✭✐✮ X ✐s ❍❛✉s❞♦r✛ ❛♥❞ A ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ X✱ ♦r ✭✐✐✮ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 28

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t f ♦❢ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X) s✉❝❤ t❤❛t s✉♣ f (A) = s✉♣ f (X) and ✐♥❢ f (A) = ✐♥❢ f (X) ✇❤❡♥❡✈❡r ✭✐✮ X ✐s ❍❛✉s❞♦r✛ ❛♥❞ A ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ X✱ ♦r ✭✐✐✮ X ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ A ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ Gδ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 29

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t f ♦❢ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X) s✉❝❤ t❤❛t s✉♣ f (A) = s✉♣ f (X) and ✐♥❢ f (A) = ✐♥❢ f (X) ✇❤❡♥❡✈❡r ✭✐✮ X ✐s ❍❛✉s❞♦r✛ ❛♥❞ A ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ X✱ ♦r ✭✐✐✮ X ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ A ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ Gδ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ A ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 30

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t f ♦❢ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X) s✉❝❤ t❤❛t s✉♣ f (A) = s✉♣ f (X) and ✐♥❢ f (A) = ✐♥❢ f (X) ✇❤❡♥❡✈❡r ✭✐✮ X ✐s ❍❛✉s❞♦r✛ ❛♥❞ A ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ X✱ ♦r ✭✐✐✮ X ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ A ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ Gδ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ A ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t

✶

❤❛s ❛♥ ❡①t❡♥s✐♦♥

✶

✱ ✇❤❡r❡ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 31

▼♦t✐✈❛t✐♦♥ ❢♦r ❇❛✐r❡ ❢✉♥❝t✐♦♥s ❑✉r❛t♦✇s❦✐ ❬✻❪ ✭✶✾✸✸✮✿ ■❢ X ✐s ❛ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ A ✐s ❛ Gδ✲s✉❜s❡t ♦❢ X✱ t❤❡♥ ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✳ ■♥ ✷✵✵✺✱ ❑❛❧❡♥❞❛ ❛♥❞ ❙♣✉r♥ý ❬✺❪ str❡♥❣t❤❡♥❡❞ ❑✉r❛r♦✇s❦✐✬s r❡s✉❧t✿ ❊✈❡r② ❡❧❡♠❡♥t f ♦❢ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X) s✉❝❤ t❤❛t s✉♣ f (A) = s✉♣ f (X) and ✐♥❢ f (A) = ✐♥❢ f (X) ✇❤❡♥❡✈❡r ✭✐✮ X ✐s ❍❛✉s❞♦r✛ ❛♥❞ A ✐s ❛ ❝♦③❡r♦✲s✉❜s❡t ♦❢ X✱ ♦r ✭✐✐✮ X ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ A ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ Gδ✲s✉❜s❡t✱ ♦r ✭✐✐✐✮ ❳ ✐s ❝♦♠♣❧❡t❡❧② r❡❣✉❧❛r ❛♥❞ A ✐s ✐ts ▲✐♥❞❡❧ö❢ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t✱ ✭♥♦ ❡①♣❧✐❝✐t ❡①t❡♥s✐♦♥ ❢♦r♠✉❧❛ ✐s ❣✐✈❡♥✮ ❛♥❞ t❤❡② ❛s❦❡❞ ❲❤❡t❤❡r ❡✈❡r② ❡❧❡♠❡♥t f ∈ B✶(A) ❤❛s ❛♥ ❡①t❡♥s✐♦♥ f ∈ B✶(X)✱ ✇❤❡r❡ A ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ❤❡r❡❞✐t❛r② ❇❛✐r❡ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X ❄

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 32

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱

❛ ♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ ✱ ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❢♦r ✭✶✮ ✇❤❡r❡ ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ ✭❞❡♣❡♥❞❡♥t ♦♥ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛ ♥♦r♠❡❞ s♣❛❝❡✱ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 33

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❢♦r ✭✶✮ ✇❤❡r❡ ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ ✭❞❡♣❡♥❞❡♥t ♦♥ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛ ♥♦r♠❡❞ s♣❛❝❡✱ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 34

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❢♦r ✭✶✮ ✇❤❡r❡ ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ ✭❞❡♣❡♥❞❡♥t ♦♥ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛ ♥♦r♠❡❞ s♣❛❝❡✱ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 35

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ f : A → Y ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❢♦r ✭✶✮ ✇❤❡r❡ ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ ✭❞❡♣❡♥❞❡♥t ♦♥ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛ ♥♦r♠❡❞ s♣❛❝❡✱ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 36

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ f : A → Y ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ (Λf ) (x) =

f (x)

❢♦r x ∈ A

t∈T f (vt)pt(x)

❢♦r x ∈ X \ A, ✭✶✮ ✇❤❡r❡ ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ ✭❞❡♣❡♥❞❡♥t ♦♥ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛ ♥♦r♠❡❞ s♣❛❝❡✱ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 37

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ f : A → Y ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ (Λf ) (x) =

f (x)

❢♦r x ∈ A

t∈T f (vt)pt(x)

❢♦r x ∈ X \ A, ✭✶✮ ✇❤❡r❡ (vt)t∈T ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ A ✭❞❡♣❡♥❞❡♥t ♦♥ x ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ f ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛ ♥♦r♠❡❞ s♣❛❝❡✱ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 38

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ f : A → Y ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ (Λf ) (x) =

f (x)

❢♦r x ∈ A

t∈T f (vt)pt(x)

❢♦r x ∈ X \ A, ✭✶✮ ✇❤❡r❡ (vt)t∈T ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ A ✭❞❡♣❡♥❞❡♥t ♦♥ x ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ f ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ C(A, Y ) ✐♥t♦ C(X, Y )✳ ▼♦r❡♦✈❡r✱ ❢♦r Y ❛ ♥♦r♠❡❞ s♣❛❝❡✱ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 39

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ f : A → Y ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ (Λf ) (x) =

f (x)

❢♦r x ∈ A

t∈T f (vt)pt(x)

❢♦r x ∈ X \ A, ✭✶✮ ✇❤❡r❡ (vt)t∈T ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ A ✭❞❡♣❡♥❞❡♥t ♦♥ x ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ f ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ C(A, Y ) ✐♥t♦ C(X, Y )✳ ▼♦r❡♦✈❡r✱ ❢♦r Y ❛ ♥♦r♠❡❞ s♣❛❝❡✱ Λ r❡str✐❝t❡❞ t♦ Cb(A, Y ) ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ Cb(X, Y )✳ ❍❡♥❝❡✱ ❢♦r ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 40

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ f : A → Y ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ (Λf ) (x) =

f (x)

❢♦r x ∈ A

t∈T f (vt)pt(x)

❢♦r x ∈ X \ A, ✭✶✮ ✇❤❡r❡ (vt)t∈T ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ A ✭❞❡♣❡♥❞❡♥t ♦♥ x ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ f ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ C(A, Y ) ✐♥t♦ C(X, Y )✳ ▼♦r❡♦✈❡r✱ ❢♦r Y ❛ ♥♦r♠❡❞ s♣❛❝❡✱ Λ r❡str✐❝t❡❞ t♦ Cb(A, Y ) ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ Cb(X, Y )✳ ❍❡♥❝❡✱ ❢♦r Y = ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ ✐♥t♦ s✉❝❤ t❤❛t r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ ✲ ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 41

❚❤❡ ❉✉❣✉♥❞❥✐ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r

✐✈❡♥ A ❛ ❝❧♦s❡❞ ✭✐♥✜♥✐t❡✮ s✉❜s❡t ♦❢ ❛ ♠❡tr✐❝ s♣❛❝❡ X✱ (pt)t∈T ❛

♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② ♦♥ X \ A✱ Y ❛ ❧♦❝❛❧❧② ❝♦♥✈❡① s♣❛❝❡✱ ❛♥❞ f : A → Y ❝♦♥t✐♥✉♦✉s✱ t❤❡ ❢♦r♠✉❧❛ (Λf ) (x) =

f (x)

❢♦r x ∈ A

t∈T f (vt)pt(x)

❢♦r x ∈ X \ A, ✭✶✮ ✇❤❡r❡ (vt)t∈T ✐s ❛ s♣❡❝✐❛❧ ❢❛♠✐❧② ✐♥ A ✭❞❡♣❡♥❞❡♥t ♦♥ x ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ f ✮✱ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❢r♦♠ C(A, Y ) ✐♥t♦ C(X, Y )✳ ▼♦r❡♦✈❡r✱ ❢♦r Y ❛ ♥♦r♠❡❞ s♣❛❝❡✱ Λ r❡str✐❝t❡❞ t♦ Cb(A, Y ) ✐s ❛♥ ✐s♦♠❡tr② ✐♥t♦ Cb(X, Y )✳ ❍❡♥❝❡✱ ❢♦r Y = ❘✱ t❛❦✐♥❣ ❧✐♠✐ts✱ ✇❡ ♦❜t❛✐♥ t❤❛t ❚❤❡ ❢♦r♠❛❧ ❢♦r♠✉❧❛ ✭✶✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Λα ❢r♦♠ Bα(A) ✐♥t♦ Bα(X) s✉❝❤ t❤❛t Λα r❡str✐❝t❡❞ t♦ t❤❡ s♣❛❝❡ Bbd

α (A) ✲ ♦❢ ❜♦✉♥❞❡❞ ❡❧❡♠❡♥ts ♦❢ Bα(A) ✲ ✐s ❛♥

✐s♦♠❡tr② ✐♥t♦ Bbd

α (X)✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 42

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t ❝❧♦s❡❞ ✐♥ ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ ✲ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t ❜❡ ❛♥ ❛♥❞ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ✱ ❛♥❞ ❧❡t

✶

✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r ✵ ❢♦r ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ ✳ P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 43

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t ❜❡ ❛♥ ❛♥❞ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ✱ ❛♥❞ ❧❡t

✶

✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r ✵ ❢♦r ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ ✳ P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 44

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t ❜❡ ❛♥ ❛♥❞ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ✱ ❛♥❞ ❧❡t

✶

✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r ✵ ❢♦r ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ ✳ P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 45

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r X ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t ❜❡ ❛♥ ❛♥❞ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ ✱ ❛♥❞ ❧❡t

✶

✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r ✵ ❢♦r ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ ✳ P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 46

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r X ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✱ ❛♥❞ ❧❡t f ∈ B✶(A)✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r ✵ ❢♦r ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ ✳ P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 47

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r X ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✱ ❛♥❞ ❧❡t f ∈ B✶(A)✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ ❢♦r ✵ ❢♦r ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ ✳ P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 48

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r X ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✱ ❛♥❞ ❧❡t f ∈ B✶(A)✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ TAf (x) =

f (x)

❢♦r x ∈ A ✵ ❢♦r x ∈ X \ A. ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② ✐♥t♦ ✳ P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 49

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r X ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✱ ❛♥❞ ❧❡t f ∈ B✶(A)✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ TAf (x) =

f (x)

❢♦r x ∈ A ✵ ❢♦r x ∈ X \ A. ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r TA : Bα(A) → Bα(X) ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② Bbd

α (A) ✐♥t♦ Bbd α (X)✳

P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 50

❆ ❣❡♥❡r❛❧ ❝❛s❡ ❚❤✉s✱ ❢♦r X ♠❡tr✐③❛❜❧❡✱ t❤❡ ❑❛❧❡♥❞❛✲❙♣✉r♥ý ♣r♦❜❧❡♠ ❤❛s ❛ ♣♦s✐t✐✈❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡ r❡q✉✐r❡♠❡♥t ✬A t♦ ❜❡ ❤❡r❡❞✐t❛r② ❇❛✐r❡✬ ✐s s✉♣❡r✢✉♦✉s✳ ❇✉t A ❝❧♦s❡❞ ✐♥ X ♠❡tr✐③❛❜❧❡ ✐s ❛ ③❡r♦ s❡t✱ ❤❡♥❝❡ ✐t ✐s ❛♥ Fσ✲Gδ s❡t✳ ❈❛♥ t❤❡ ❛❜♦✈❡ r❡s✉❧t ❜❡ ❡①t❡♥❞❡❞ ❢♦r s✉❝❤ s✉❜s❡ts❄ ❨❡s✱ ❡✈❡♥ ❢♦r X ♥♦r♠❛❧ ✇✐t❤ ❛ s✐♠♣❧❡r ❢♦r♠ ♦❢ ❧✐♥❡❛r ❡①t❡♥s✐♦♥s✳ ❚❤❡♦r❡♠ ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✱ ❛♥❞ ❧❡t f ∈ B✶(A)✳ ❚❤❡♥ t❤❡ ❢♦r♠✉❧❛ TAf (x) =

f (x)

❢♦r x ∈ A ✵ ❢♦r x ∈ X \ A. ✭✷✮ ❞❡✜♥❡s ❛ ♣♦s✐t✐✈❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r TA : Bα(A) → Bα(X) ✇❤✐❝❤ ♠❛♣s ✐s♦♠❡tr✐❝❛❧❧② Bbd

α (A) ✐♥t♦ Bbd α (X)✳

P❢✳ ❇② ♠❡❛♥s ♦❢ t❤❡ ❚✐❡t③❡ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 51

❈♦r♦❧❧❛r② ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✳ ❚❤❡♥ ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ ✳ ❙✐♠✐❧❛r❧②✱ ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 52

❈♦r♦❧❧❛r② ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✳ ❚❤❡♥ Bα(X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ Bα(A) × Bα(A′)✳ ❙✐♠✐❧❛r❧②✱ ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ ✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 53

❈♦r♦❧❧❛r② ✶✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✳ ❚❤❡♥ Bα(X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ Bα(A) × Bα(A′)✳ ❙✐♠✐❧❛r❧②✱ Bbd

α (X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ Bbd α (A) × Bbd α (A′)✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 54

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ ✲ s✉❝❤ t❤❛t

✶

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ ❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ ♦♥ ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿

✶

✇❤❡r❡

✶

❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t ♦❢ t❤❡ s❡t ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ ✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ ❛r❡ ❞❡♥♦t❡❞ ❜② ❛♥❞

✶

✱ r❡s♣❡❝t✐✈❡❧②✳ ■t ✐s ❦♥♦✇♥ t❤❛t

✶

❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱ ❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 55

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ ♦♥ ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿

✶

✇❤❡r❡

✶

❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t ♦❢ t❤❡ s❡t ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ ✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ ❛r❡ ❞❡♥♦t❡❞ ❜② ❛♥❞

✶

✱ r❡s♣❡❝t✐✈❡❧②✳ ■t ✐s ❦♥♦✇♥ t❤❛t

✶

❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱ ❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 56

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ ♦♥ ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿

✶

✇❤❡r❡

✶

❛♥❞ ✐s ❝♦♥t✐♥✉♦✉s ❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t ♦❢ t❤❡ s❡t ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ ✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ ❛r❡ ❞❡♥♦t❡❞ ❜② ❛♥❞

✶

✱ r❡s♣❡❝t✐✈❡❧②✳ ■t ✐s ❦♥♦✇♥ t❤❛t

✶

❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱ ❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 57

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t ♦❢ t❤❡ s❡t ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ ✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ ❛r❡ ❞❡♥♦t❡❞ ❜② ❛♥❞

✶

✱ r❡s♣❡❝t✐✈❡❧②✳ ■t ✐s ❦♥♦✇♥ t❤❛t

✶

❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱ ❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 58

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t D ♦❢ X t❤❡ s❡t C(f|D) ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ f|F ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ D✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ ❛r❡ ❞❡♥♦t❡❞ ❜② ❛♥❞

✶

✱ r❡s♣❡❝t✐✈❡❧②✳ ■t ✐s ❦♥♦✇♥ t❤❛t

✶

❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱ ❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 59

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t D ♦❢ X t❤❡ s❡t C(f|D) ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ f|F ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ D✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ ❛r❡ ❞❡♥♦t❡❞ ❜② ❛♥❞

✶

✱ r❡s♣❡❝t✐✈❡❧②✳ ■t ✐s ❦♥♦✇♥ t❤❛t

✶

❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱ ❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 60

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t D ♦❢ X t❤❡ s❡t C(f|D) ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ f|F ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ D✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ X ❛r❡ ❞❡♥♦t❡❞ ❜② P(X) ❛♥❞ B∗

✶(X)✱ r❡s♣❡❝t✐✈❡❧②✳

■t ✐s ❦♥♦✇♥ t❤❛t

✶

❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱ ❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 61

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t D ♦❢ X t❤❡ s❡t C(f|D) ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ f|F ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ D✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ X ❛r❡ ❞❡♥♦t❡❞ ❜② P(X) ❛♥❞ B∗

✶(X)✱ r❡s♣❡❝t✐✈❡❧②✳

■t ✐s ❦♥♦✇♥ t❤❛t P(X) = B∗

✶(X) ❢♦r X ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱

❛♥❞ t❤❛t ✐✛ ✐s ❛ ✲s♣❛❝❡ ✭❂ ❡✈❡r② ✲s✉❜s❡t ♦❢ ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 62

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t D ♦❢ X t❤❡ s❡t C(f|D) ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ f|F ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ D✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ X ❛r❡ ❞❡♥♦t❡❞ ❜② P(X) ❛♥❞ B∗

✶(X)✱ r❡s♣❡❝t✐✈❡❧②✳

■t ✐s ❦♥♦✇♥ t❤❛t P(X) = B∗

✶(X) ❢♦r X ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱

❛♥❞ t❤❛t P(X) = C(X) ✐✛ X ✐s ❛ P✲s♣❛❝❡ ✭❂ ❡✈❡r② Gδ✲s✉❜s❡t ♦❢ X ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡

✶

✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 63

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t D ♦❢ X t❤❡ s❡t C(f|D) ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ f|F ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ D✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ X ❛r❡ ❞❡♥♦t❡❞ ❜② P(X) ❛♥❞ B∗

✶(X)✱ r❡s♣❡❝t✐✈❡❧②✳

■t ✐s ❦♥♦✇♥ t❤❛t P(X) = B∗

✶(X) ❢♦r X ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱

❛♥❞ t❤❛t P(X) = C(X) ✐✛ X ✐s ❛ P✲s♣❛❝❡ ✭❂ ❡✈❡r② Gδ✲s✉❜s❡t ♦❢ X ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡ P(X) ⊂ B✶(X)✱ ❛♥❞ ✐❢ ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t

✶

✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 64

❚❤❡ ❝❛s❡ ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s X ✐s ❛ ❍❛✉s❞♦r✛ s♣❛❝❡✳ ❆ ❢✉♥❝t✐♦♥ f : X → ❘ ✐s s❛✐❞ t♦ ❜❡ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ (Xn(f )) ♦❢ ❝❧♦s❡❞ s✉❜s❡ts ♦❢ X ✲ ❞❡♣❡♥❞✐♥❣ ♦♥ f ✲ s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f )

❛♥❞ t❤❡ r❡str✐❝t✐♦♥s f|Xn(f ) ❛r❡ ❝♦♥t✐♥✉♦✉s ❢♦r ❛❧❧ n❀ ❡✈❡r② ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥ f ♦♥ X ✐s ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s✿ X = ∞

n=✶ An(f ) ✇❤❡r❡ An(f ) = f −✶[−n, n] ❛♥❞ f|An(f ) ✐s

❝♦♥t✐♥✉♦✉s ∀n❀ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥ ✐❢ ❢♦r ❡✈❡r② ❝❧♦s❡❞ s✉❜s❡t D ♦❢ X t❤❡ s❡t C(f|D) ✲ ♦❢ ❝♦♥t✐♥✉♦✉s ♣♦✐♥ts ♦❢ f|F ✲ ❤❛s ♥♦♥❡♠♣t② ✐♥t❡r✐♦r ✭✐♥ t❤❡ ✐♥❞✉❝❡❞ t♦♣♦❧♦❣② ♦♥ D✮✳ ❚❤❡ ❧✐♥❡❛r s♣❛❝❡s ♦❢ ♣✐❡❝❡✇✐s❡ ❝♦♥t✐♥✉♦✉s ❛♥❞ ❇❛✐r❡✲♦♥❡✲st❛r ❢✉♥❝t✐♦♥s ♦♥ X ❛r❡ ❞❡♥♦t❡❞ ❜② P(X) ❛♥❞ B∗

✶(X)✱ r❡s♣❡❝t✐✈❡❧②✳

■t ✐s ❦♥♦✇♥ t❤❛t P(X) = B∗

✶(X) ❢♦r X ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✱

❛♥❞ t❤❛t P(X) = C(X) ✐✛ X ✐s ❛ P✲s♣❛❝❡ ✭❂ ❡✈❡r② Gδ✲s✉❜s❡t ♦❢ X ✐s ♦♣❡♥✮ ❬✾❪✳ ❲❡ ❛❧s♦ ❤❛✈❡ P(X) ⊂ B✶(X)✱ ❛♥❞ ✐❢ X ✐s ❛♥ ✐♥❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t P(X) = B✶(X)✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 65

❚❤❡♦r❡♠ ✷✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✳ ❚❤❡♥ t❤❡ ❧✐❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r TA ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✶ ♠❛♣s P(A) ✐♥t♦ P(X)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ ✱ ✇❤❡♥❝❡

✶

✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦

✶ ✶

✱ ❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✳ P❢✳ ❇② ❚✐❡t③❡✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 66

❚❤❡♦r❡♠ ✷✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✳ ❚❤❡♥ t❤❡ ❧✐❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r TA ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✶ ♠❛♣s P(A) ✐♥t♦ P(X)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ P(X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ P(A) × P(A′)✱ ✇❤❡♥❝❡

✶

✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦

✶ ✶

✱ ❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✳ P❢✳ ❇② ❚✐❡t③❡✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 67

❚❤❡♦r❡♠ ✷✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✳ ❚❤❡♥ t❤❡ ❧✐❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r TA ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✶ ♠❛♣s P(A) ✐♥t♦ P(X)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ P(X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ P(A) × P(A′)✱ ✇❤❡♥❝❡ B∗

✶(X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ B∗ ✶(A) × B∗ ✶(A′)✱ ❢♦r X ❛ ❝♦♠♣❧❡t❡

♠❡tr✐❝ s♣❛❝❡✳ P❢✳ ❇② ❚✐❡t③❡✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 68

❚❤❡♦r❡♠ ✷✳ ▲❡t A ❜❡ ❛♥ Fσ ❛♥❞ Gδ s✉❜s❡t ♦❢ ❛ ♥♦r♠❛❧ s♣❛❝❡ X✳ ❚❤❡♥ t❤❡ ❧✐❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r TA ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✶ ♠❛♣s P(A) ✐♥t♦ P(X)✳ ■♥ ♣❛rt✐❝✉❧❛r✱ P(X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ P(A) × P(A′)✱ ✇❤❡♥❝❡ B∗

✶(X) ✐s ♦r❞❡r ✐s♦♠♦r♣❤✐❝ t♦ B∗ ✶(A) × B∗ ✶(A′)✱ ❢♦r X ❛ ❝♦♠♣❧❡t❡

♠❡tr✐❝ s♣❛❝❡✳ P❢✳ ❇② ❚✐❡t③❡✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 69

P✵(X) ❞❡♥♦t❡s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❡❧❡♠❡♥ts f ∈ P(X) s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f ) ❛♥❞ ❡❛❝❤ ✭❝❧♦s❡❞✮ Xn(f ) ✐s Gδ✳

■❢ ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t❤❡♥

✵

✳ ▼♦r❡♦✈❡r✱ ✐❢ ✐s ♥♦r♠❛❧✱ t❤❡ s❡t ♦❢ ❛❧❧ ♥♦♥✲♥❡❣❛t✐✈❡ ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥s ♦♥ ✐s ❛ ❣❡♥❡r❛t✐♥❣ ❝♦♥❡ ♦❢

✵

✿

✵

✭✸✮ ❚❤❡♦r❡♠ ✸✳ ▲❡t ❜❡ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❛♥❞ ❛ ❝❧♦s❡❞ s✉❜s❡t ♦❢ ✳ ❚❤❡♥ ♠❛♣s

✵

✐♥t♦

✵

✳ P❢✳ ❇② t❤❡ ✉s❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✸✮✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 70

P✵(X) ❞❡♥♦t❡s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❡❧❡♠❡♥ts f ∈ P(X) s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f ) ❛♥❞ ❡❛❝❤ ✭❝❧♦s❡❞✮ Xn(f ) ✐s Gδ✳

■❢ X ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t❤❡♥ P✵(X) = P(X)✳ ▼♦r❡♦✈❡r✱ ✐❢ ✐s ♥♦r♠❛❧✱ t❤❡ s❡t ♦❢ ❛❧❧ ♥♦♥✲♥❡❣❛t✐✈❡ ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥s ♦♥ ✐s ❛ ❣❡♥❡r❛t✐♥❣ ❝♦♥❡ ♦❢

✵

✿

✵

✭✸✮ ❚❤❡♦r❡♠ ✸✳ ▲❡t ❜❡ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❛♥❞ ❛ ❝❧♦s❡❞ s✉❜s❡t ♦❢ ✳ ❚❤❡♥ ♠❛♣s

✵

✐♥t♦

✵

✳ P❢✳ ❇② t❤❡ ✉s❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✸✮✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 71

P✵(X) ❞❡♥♦t❡s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❡❧❡♠❡♥ts f ∈ P(X) s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f ) ❛♥❞ ❡❛❝❤ ✭❝❧♦s❡❞✮ Xn(f ) ✐s Gδ✳

■❢ X ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t❤❡♥ P✵(X) = P(X)✳ ▼♦r❡♦✈❡r✱ ✐❢ X ✐s ♥♦r♠❛❧✱ t❤❡ s❡t U+(X) ♦❢ ❛❧❧ ♥♦♥✲♥❡❣❛t✐✈❡ ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥s ♦♥ X ✐s ❛ ❣❡♥❡r❛t✐♥❣ ❝♦♥❡ ♦❢ P✵(X)✿

✵

✭✸✮ ❚❤❡♦r❡♠ ✸✳ ▲❡t ❜❡ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❛♥❞ ❛ ❝❧♦s❡❞ s✉❜s❡t ♦❢ ✳ ❚❤❡♥ ♠❛♣s

✵

✐♥t♦

✵

✳ P❢✳ ❇② t❤❡ ✉s❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✸✮✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 72

P✵(X) ❞❡♥♦t❡s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❡❧❡♠❡♥ts f ∈ P(X) s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f ) ❛♥❞ ❡❛❝❤ ✭❝❧♦s❡❞✮ Xn(f ) ✐s Gδ✳

■❢ X ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t❤❡♥ P✵(X) = P(X)✳ ▼♦r❡♦✈❡r✱ ✐❢ X ✐s ♥♦r♠❛❧✱ t❤❡ s❡t U+(X) ♦❢ ❛❧❧ ♥♦♥✲♥❡❣❛t✐✈❡ ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥s ♦♥ X ✐s ❛ ❣❡♥❡r❛t✐♥❣ ❝♦♥❡ ♦❢ P✵(X)✿ P✵(X) = U+(X) − U+(X). ✭✸✮ ❚❤❡♦r❡♠ ✸✳ ▲❡t ❜❡ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❛♥❞ ❛ ❝❧♦s❡❞ s✉❜s❡t ♦❢ ✳ ❚❤❡♥ ♠❛♣s

✵

✐♥t♦

✵

✳ P❢✳ ❇② t❤❡ ✉s❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✸✮✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 73

P✵(X) ❞❡♥♦t❡s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❡❧❡♠❡♥ts f ∈ P(X) s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f ) ❛♥❞ ❡❛❝❤ ✭❝❧♦s❡❞✮ Xn(f ) ✐s Gδ✳

■❢ X ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t❤❡♥ P✵(X) = P(X)✳ ▼♦r❡♦✈❡r✱ ✐❢ X ✐s ♥♦r♠❛❧✱ t❤❡ s❡t U+(X) ♦❢ ❛❧❧ ♥♦♥✲♥❡❣❛t✐✈❡ ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥s ♦♥ X ✐s ❛ ❣❡♥❡r❛t✐♥❣ ❝♦♥❡ ♦❢ P✵(X)✿ P✵(X) = U+(X) − U+(X). ✭✸✮ ❚❤❡♦r❡♠ ✸✳ ▲❡t X ❜❡ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❛♥❞ A ❛ ❝❧♦s❡❞ Gδ s✉❜s❡t ♦❢ X✳ ❚❤❡♥ TA ♠❛♣s P✵(A) ✐♥t♦ P✵(X)✳ P❢✳ ❇② t❤❡ ✉s❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✸✮✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 74

P✵(X) ❞❡♥♦t❡s t❤❡ s✉❜s♣❛❝❡ ♦❢ ❡❧❡♠❡♥ts f ∈ P(X) s✉❝❤ t❤❛t X = ∞

n=✶ Xn(f ) ❛♥❞ ❡❛❝❤ ✭❝❧♦s❡❞✮ Xn(f ) ✐s Gδ✳

■❢ X ✐s ♣❡r❢❡❝t❧② ♥♦r♠❛❧ t❤❡♥ P✵(X) = P(X)✳ ▼♦r❡♦✈❡r✱ ✐❢ X ✐s ♥♦r♠❛❧✱ t❤❡ s❡t U+(X) ♦❢ ❛❧❧ ♥♦♥✲♥❡❣❛t✐✈❡ ❝❧♦s❡❞ ❣r❛♣❤ ❢✉♥❝t✐♦♥s ♦♥ X ✐s ❛ ❣❡♥❡r❛t✐♥❣ ❝♦♥❡ ♦❢ P✵(X)✿ P✵(X) = U+(X) − U+(X). ✭✸✮ ❚❤❡♦r❡♠ ✸✳ ▲❡t X ❜❡ ❛ ♥♦r♠❛❧ s♣❛❝❡ ❛♥❞ A ❛ ❝❧♦s❡❞ Gδ s✉❜s❡t ♦❢ X✳ ❚❤❡♥ TA ♠❛♣s P✵(A) ✐♥t♦ P✵(X)✳ P❢✳ ❇② t❤❡ ✉s❡ ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ✭✸✮✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 75

❘✳ ▼✳ ❆r♦♥✱ P✳ ❉✳ ❇❡r♥❡r✱ ❆ ❍❛❤♥✲❇❛♥❛❝❤ ❡①t❡♥s✐♦♥ t❤❡♦r❡♠ ❢♦r ❛♥❛❧②t✐❝ ♠❛♣♣✐♥❣s✱ ❇✉❧❧✳ ❙♦❝✳ ▼❛t❤✳ ❋r❛♥❝❡ ✶✵✻ ✭✶✾✼✽✮✱ ♥♦✳ ✶✱ ✸✕✷✹✳ ❙✳ ❇r♦♠❜❡r❣✱ ❆♥ ❡①t❡♥s✐♦♥ ✐♥ ❝❧❛ss C ✶✱ ❇♦❧✳ ❙♦❝✳ ▼❛t✳ ▼❡①✳ ■■✱ ❙❡r✳ ✷✼ ✭✶✾✽✷✮✱ ✸✺✕✹✹✳ ❏✳ ❉✉❣✉♥❞❥✐✱ ❆♥ ❡①t❡♥s✐♦♥ ♦❢ ❚✐❡t③❡✬s t❤❡♦r❡♠✱ P❛❝✐✜❝ ❏✳ ▼❛t❤✳ ✶ ✭✶✾✺✶✮✱ ✸✺✸✕✸✻✼✳ ❈✳ ▲✳ ❋❡✛❡r♠❛♥✱ C m ❡①t❡♥s✐♦♥ ❜② ❧✐♥❡❛r ♦♣❡r❛t♦rs✱ ❆♥♥❛❧s ♦❢ ▼❛t❤✳ ✈♦❧✳ ✶✻✻ ✭✷✵✵✼✮✱ ◆♦✳ ✸✱ ✼✼✾✕✽✸✺✳ ❖✳ ❋✳ ❑✳ ❑❛❧❡♥❞❛✱ ❏✳ ❙♣✉r♥ý✱ ❊①t❡♥❞✐♥❣ ❇❛✐r❡✲♦♥❡ ❢✉♥❝t✐♦♥s ♦♥ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳ ✶✹✾ ✭✷✵✵✺✮✱ ♥♦✳ ✶✲✸✱ ✶✾✺✕✷✶✻✳ ❑✳ ❑✉r❛t♦✇s❦✐✱ ❙✉r ❧❡s t❤é♦ré♠❡s t♦♣♦❧♦❣✐q✉❡s ❞❡ ❧❛ t❤é♦r✐❡ ❞❡s ❢♦♥❝t✐♦♥s ❞❡ ✈❛r✐❛❜❧❡s ré❡❧❧❡s✱ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s ✶✾✼ ✭✶✾✸✸✮✱ ✶✾✕✷✵✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s

SLIDE 76

❏✳ ▼❡rr✐❡♥✱ Pr♦❧♦♥❣❛t❡✉rs ❞❡ ❢♦♥❝✐♦♥s ❞✐✛❡r❡♥t✐❛❜❧❡s ❞✉♥❡ ✈❛r✐❛❜❧❡ r❡❧❧❡✱ ❏✳ ▼❛t❤✳ P✉r❡s ❆♣♣❧✳ ✭✾✮ ✹✺✱ ✭✶✾✻✻✮✱ ✷✾✶✕✸✵✾✳ ❍✳ ❘✳ ❙❤❛t❡r②✱ ❏✳ ❩❛❢❛r❛♥✐✱ ❚❤❡ ❡q✉❛❧✐t② ❜❡t✇❡❡♥ ❇♦r❡❧ ❛♥❞ ❇❛✐r❡ ❝❧❛ss❡s✱ ❘❡❛❧ ❆♥❛❧✳ ❊①❝❤✳ ✸✵✭✶✮✱ ✭✷✵✵✹✴✷✵✵✺✮✱ ✸✼✸✕✸✽✹✳ ▼✳ ❲ó❥t♦✇✐❝③✱ ❲✳ ❙✐❡❣✱ P✲s♣❛❝❡s ❛♥❞ ❛♥ ✉♥❝♦♥❞✐t✐♦♥❛❧ ❝❧♦s❡❞ ❣r❛♣❤ t❤❡♦r❡♠✱ ❘❆❈❙❆▼ ✶✵✹ ✭✶✮✱ ✭✷✵✶✵✮✱ ✶✸✕✶✽✳ ▼✳ ❲ó❥t♦✇✐❝③✱ ❲✳ ❙✐❡❣✱ ❆✣♥❡ ❡①t❡♥s✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s ✇✐t❤ ❛ ❝❧♦s❡❞ ❣r❛♣❤✱ ❖♣✉s❝✉❧❛ ▼❛t❤✳ ✸✺✱ ♥♦✳ ✻ ✭✷✵✶✺✮✱ ✾✼✸✕✾✼✽✳

▼❛r❡❦ ❲ó❥t♦✇✐❝③

✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s