S is sometimes called a set function . called a (positive) measure if - - PowerPoint PPT Presentation

• Defjnition. Let (S, S) be a measurable space. A mapping µ : S → [0, ∞] is

called a (positive) measure if

• 1. µ(∅) = 0, and
• 2. µ(∪nAn) = ∑

n∈N µ(An), for all pairwise disjoint {An}n∈N in S.

A triple (S, S, µ) consisting of a non-empty set, a σ-algebra S on it and a measure µ on S is called a measure space.

• A mapping whose domain is some nonempty set A of subsets of some set

S is sometimes called a set function.

• If 2. above is required only for fjnite sumes, µ is called fjnitely-additive

measure.

A measure µ on the measurable space (S, S) is called

• 1. a probability measure, if µ(S) = 1,
• 2. a fjnite measure, if µ(S) < ∞,
• 3. a σ-fjnite measure, if there exists a sequence {An}n∈N in S such that

∪nAn = S and µ(An) < ∞,

• 4. difguse or atom-free, if µ({x}) = 0, whenever x ∈ S and {x} ∈ S.

A set N ∈ S is said to be null if µ(N) = 0.

• 1. Measures on countable sets. Suppose that S is a fjnite or countable
• set. Then each measure µ on S = 2S is of the form

µ(A) = ∑

x∈A

p(x), for some function p : S → [0, ∞]. In particular, for a fjnite set S with N elements, if p(x) = 1/N then µ is a probability measure called the uniform measure on S.

• 2. Dirac measure. For x ∈ S, we defjne the set function δx on S by

δx(A) = { 1, x ∈ A, 0, x ̸∈ A.

• 3. The counting measure. Defjne the counting measure

µ : S → [0, ∞] by µ(A) = { #A, A is fjnite, ∞, A is infjnite..

Let (S, S, µ) be a measure space.

• 1. For A1, . . . , An ∈ S with Ai ∩ Aj = ∅, for i ̸= j, we have

n

∑

i=1

µ(Ai) = µ(∪n

i=1Ai)

(Finite additivity)

• 2. If A, B ∈ S, A ⊆ B, then

µ(A) ≤ µ(B) (Monotonicity of measures)

• 3. If {An}n∈N in S is increasing, then

µ(∪nAn) = lim

n µ(An) = sup n µ(An).

(Continuity with respect to increasing sequences)

• 4. If {An}n∈N in S is decreasing and µ(A1) < ∞, then

µ(∩nAn) = lim

n µ(An) = inf n µ(An).

(Continuity with respect to decreasing sequences)

• 5. For a sequence {An}n∈N in S, we have

µ(∪nAn) ≤ ∑

n∈N

µ(An). (Subadditivity)

Let {An}n∈N be a sequence of subsets of S. We defjne lim infn An by lim inf

n

An = ∪nBn, where Bn = ∩k≥nAk, is called the limit inferior of the sequence An. It is also denoted by limnAn

• r {An, ev.} (ev. stands for eventually).

Similarly, the subset lim supn An of S, defjned by lim sup

n

An = ∩nBn, where Bn = ∪k≥nAk, is called the limit superior of the sequence An. It is also denoted by limnAn

• r {An, i.o.} (i.o. stands for infjnitely often).
• Problem. Let An = {0, 1

n, 2 n, . . . , n2 n }. Find lim sup An, lim inf An. How

about if An = {0, 1

2n , 2 2n , . . . , 4n 2n }?

Remember lim supn An = ∩n ∪k≥n Ak.

• Proposition. (The Borel-Cantelli Lemma) Let (S, S, µ) be a measure space,

and let {An}n∈N be a sequence of sets in S with the property that ∑

n∈N µ(An) <

∞. Then µ(lim sup

n

An) = 0.

Remember, product cylinders on {−1, 1}N are Cn1,...,nk;b1,...,bk = { s = (s1, s2, . . . ) ∈ {−1, 1}N : sn1 = b1, . . . , snk = bk } , for some k ∈ N, and a choice of 1 ≤ n1 < n2 < · · · < nk ∈ N of coordinates and the corresponding values b1, b2, . . . , bk ∈ {−1, 1}. In the language of elementary probability, each cylinder corresponds to the event when the outcome of the ni-th coin is bi ∈ {−1, 1}, for i = 1, . . . , n. The measure (probability) of this event can only be given by µC(Cn1,...,nk;b1,...,bk) = 1

2 × 1 2 × · · · × 1 2

• k times

= 2−k. (1) The hard part is to extend this defjnition to all elements of S, and not only cylinders.

• Theorem. (Caratheodory’s Extension Theorem) Let S be a non-empty set,

let A be an algebra of its subsets and let µ : A → [0, ∞] be a set-function with the following properties:

• 1. µ(∅) = 0, and
• 2. µ(A) = ∑∞

n=1 µ(An), if {An}n∈N ⊆ A is a partition of A.

• Proposition. (Existence of the coin-toss measure) There exists a measure

µC on ({−1, 1}N, S) with the “obvious” values on the cylinders.

• Theorem. (Dynkin) Let P be a π-system on a non-empty set S, and let Λ

be a λ-system which contains P. Then Λ also contains the σ-algebra σ(P) generated by P.

• Proposition. Let (S, S) be a measurable space, and let P be a π-system

which generates S. Suppose that µ1 and µ2 are two measures on S with the property that µ1(S) = µ2(S) < ∞ and µ1(A) = µ2(A), for all A ∈ P. Then µ1 = µ2, i.e., µ1(A) = µ2(A), for all A ∈ S.

• Proposition. The measure µC is the unique measure on ({−1, 1}N, S) with

the “obvious” values on the cylinders.

• Defjnition. Let (S, S, µ) be a measure space and let (T, T ) be a measurable
• space. The measure f∗µ on (T, T ), defjned by

f∗µ(B) = µ(f−1(B)), for B ∈ T , is called the push-forward of the measure µ by f. Let f : {−1, 1}N → [0, 1] be the mapping given by f(s) =

∞

∑

k=1

(

1+sk 2

) 2−k, s ∈ {−1, 1}N. The continuity of f implies that f : ({−1, 1}N, S) → ([0, 1], B([0, 1])) is a measurable mapping. Therefore, the push-forward λ = f∗(µ) is well defjned

• n ([0, 1], B([0, 1])), and we call it the Lebesgue measure on [0, 1].

• Proposition. The Lebesgue measure λ on ([0, 1], B([0, 1])) satisfjes

λ([a, b)) = b − a, λ({a}) = 0 for 0 ≤ a < b ≤ 1.

• Problem. Show that the Lebesgue measure is translation invariant. More

precisely, for B ∈ B([0, 1]) and x ∈ [0, 1), we have

• 1. B +1 x = {b + x (mod 1) : b ∈ B} is in B([0, 1]) and
• 2. λ(B +1 x) = λ(B),

where, for a ∈ [0, 2), we defjne a (mod 1) = a − 1{a>1}.

For a general B ∈ B(R), we set λ(B) = ∑∞

n=−∞ λ

(( B ∩ [n, n + 1) ) − n ) .

• Problem. Let λ be the Lebesgue measure on (R, B(R)). Show that
• 1. λ([a, b)) = b − a, λ({a}) = 0 for a < b,
• 2. λ is σ-fjnite but not fjnite,
• 3. λ(B + x) = λ(B), for all B ∈ B(R) and x ∈ R.