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Lectures on Measure and Integration
By Harold Widom Dover Publications, Inc.
Copyright © 2016 Harold Widom
All rights reserved.
ISBN: 978-0-486-81659-3
CHAPTER 1
MEASURES
Definition. Let S be a set and Σ a collection of subsets of S. Σ is said to be a field if:
(i) [empty set] [member of] ([empty set] denotes the empty set);
(ii) A [member of] Σ implies A' [member of] Σ (A' denotes the complement of A);
(iii) A [member of] Σ, B [member of] Σ implies A [union] B [member of] Σ.
Thus Σ contains the empty set, the complement of each of its members, and finite unions of its members. A field Σ is sometimes also referred to as an algebra of sets.
Definition. An additive set function μ on Σ is a function with domain Σ such that
μ(A [union] B) = μ(A) + μ(B) if A [intersection] B = [empty set].
The range can be any set which has a binary operation (group, field, vector space); however we shall consider only the real numbers, complex numbers, and the non-negative reals with ∞ adjoined (the extended non-negative reals, henceforth denoted by R*) as ranges of set functions.
Some trivial properties of an additive set function μ: [SIGMA ->R* are
(i) μ([empty set]) = 0 unless μ is identically ∞;
(ii) A [subset] B implies μ(A) ≤ μ(B);
(iii) if A [contains] and μ(B) < ∞, then
μ(A - B) = μ(A) - μ(B);
(iv) Ai [member of] Σ, i = 1, ..., n implies
[MATHEMATICAL EXPRESSION OMITTED]
Proof of (iv). We have
[MATHEMATICAL EXPRESSION OMITTED]
Thus, since the right side is now a disjoint union,
[MATHEMATICAL EXPRESSION OMITTED]
and so by (ii)
[MATHEMATICAL EXPRESSION OMITTED]
Examples: (i) The finite unions of half open intervals
[a, b) [subset] [A, B)
form a field Σ of subsets of [A, B) (finite unions of closed or open subsets of an interval do not form a field). Let φ be a non-decreasing function on [A, B) and define
μφ([a, b)) = φ(b) - φ(a).
μφ gives rise, in the obvious manner, to an additive set function on Σ. Of particular interest is the case φ(x) = x, which will later lead to the familiar "dx" in integration.
(ii) Let S be any set and Σ be the collection of all subsets of S. Define μ: Σ ->Z+[union] {∞} by letting μ(A) be the number of elements in A (here Z+ denotes the non-negative integers). This μ is called the counting measure.
Definitions. A field Σ is called a σ-field if Ai [member of] Σ, i = 1, 2, ... implies [union]∞i=1Ai [member of] Σ. An additive set function μ defined on a field Σ, and whose range is also a topological space, is called σ-additive or countably additive on Σ if when the Ai are pairwise disjoint and [union]∞i=1Ai is in Σ then Σ∞i=1 μ(Ai) is defined and
[MATHEMATICAL EXPRESSION OMITTED]
In any space S there are at least two σ-fields, namely those consisting of all subsets of S and of S and φ only.
Definition. An R*-valued countably additive set function on a field Σ is called a measure on Σ.
Consider the non-negative additive set function μφ defined previously. We have
[MATHEMATICAL EXPRESSION OMITTED]
where δ is so chosen that b - δ/2 >a. If μφ is to be countably additive, then we must have
[MATHEMATICAL EXPRESSION OMITTED]
implying that φ(b) = φ(b-) and so φ must be left continuous. This condition is also sufficient for the countable additivity of μφ. However we may remove the restriction that φ be left continuous by changing our definition of μφ to read
μφ([a, b)) = φ(b-) - φ(a-)
then μφ is a measure for all non-decreasing functions φ.
Definition. Let μ be an additive R*-valued function on a field Σ of subsets of a topological space S. μ is said to be regular if given ε > 0 and A [member of] Σ, we can find E, F [member of] Σ with [bar.E] compact, [bar.E] [subset] A [subset] F0, μ(A - E) < ε, and μ[(F - A) < ε. (Here Fdenotes the interior of F0 denotes the interior of F.)
Example. The μφ are regular:
From the definition of μφ it suffices to consider only sets of the field Σ of the form A = [a, b). Let F = [a - δ, b), E = [a, b - δ), where the appropriate δ > 0 is to be chosen.
μφ(A - E) = μφ([b - δ, b)) = φ(b-) - φ(b - δ-)
μφ(F - A) = μφ([a - δ, a)) = φ(a-) - φ(a - δ-).
Since [MATHEMATICAL EXPRESSION OMITTED] and [MATHEMATICAL EXPRESSION OMITTED] we may pick δ > 0 sufficiently small so that μφ(A - E) < ε and μφ(F - A) < ε.
Theorem 1. If μ is a regular additive set function on a field Σ, then μ is a measure (i.e., μ is countably additive).
Proof. Let Ai [member of] Σ be pairwise disjoint with [union]∞i=1Ai [member of] Σ. For each n.
[MATHEMATICAL EXPRESSION OMITTED]
whence
[MATHEMATICAL EXPRESSION OMITTED]
Now we must get this inequality going in the other direction. Let E [member of] Σ, [bar.E] compact, with [bar.E] [subset] [union]∞i=1Ai and μ([union]∞i=1Ai - E) < ε. Choose Fi [member of] Σ, such that F0i [contains] Ai, μ(Fi - Ai) < ε/2i. The F0i cover [bar.E] so that for some M, we have [union]Mi=1Fi [contains] E. Then
[MATHEMATICAL EXPRESSION OMITTED]
Since ε > 0 was arbitrary we have
[MATHEMATICAL EXPRESSION OMITTED]
and the theorem is proved.
Theorem 2. Let Ω be a collection of subsets of a set S. There exists a unique smallest σ-field of subsets of S containing Ω. (We call this the σ-field generated by Ω and write this as σ(Ω)).
Proof. Let Σ be the intersection of all the σ-fields containing Ω. (We know there is at least one.) Since the intersection of σ-fields is a σ]-field, Σ is the required σ-field.
Given a measure μ on a field Σ of subsets of a set S, the question of extending μ to a measure on the σ-field σ(Σ) generated by Σ naturally arises. The answer is that we can in fact extend μ to a measure on a σ-field containing σ(Σ). Our technique of constructing this extension will be to give a postulational description of what is called an outer measure function. In general this will not be a measure function, but will have for its domain the class of all subsets of S. We shall prove that a suitable restriction of an outer measure function to a smaller domain, which is however a σ-field, always yields a measure function. Our goal thus becomes to construct an outer measure on all subsets of S if we are given a measure on some field of subsets of S.
Definition. Let λ: [2S ->R*, where 2S denotes the collection of all subsets of S. We say that λ is an outer measure if the following are satisfied:
(i) λ([empty set]) = 0;
(ii) if A [subset] B, then λ(A) ≤ λ(B);
(iii) λ([union]∞i=1Ai) ≤ Σ∞i=1 λ(Ai) for any sets A1, A2, ... [member of] S.
It is a simple exercise to show that an outer measure which is finitely additive is also countably additive (use (iii) and an inequality derived at the beginning of the proof of Theorem 1).
Theorem 3. Let μ be a measure which is defined on a field Σ of S. Define μ*: 2S ->R* by
[MATHEMATICAL EXPRESSION OMITTED]
where the inf is taken over all sequences {Ai} Σ satisfying [union]∞i=1Ai [contains] E. Then μ* is an outer measure and μ* = μ on Σ.
Proof. We first show that μ* = μ on Σ. For A [member of] Σ, clearly
(1) μ*(A) ≤ μ(A).
By letting B1 = A1, and
[MATHEMATICAL EXPRESSION OMITTED]
we have [union]∞i=1Ai = [union]∞i=1Bi with the Bi's disjoint, whence
[MATHEMATICAL EXPRESSION OMITTED]
Therefore
[MATHEMATICAL EXPRESSION OMITTED]
and by using (1) we have μ(A) = μ*(A).
For the first statement of the theorem we prove that properties (i)-(iii) of outer measures belong to μ*.
(i) μ*([empty set]) = μ([empty set]) = 0 since [empty set] [member of] Σ;
(ii) Let A [subset] B, A, B [member of] 2S and let {Bj} [subset] Σ be such that [union]∞j=1Bj [contains] B and
[MATHEMATICAL EXPRESSION OMITTED]
But [union]∞j=1Bj [contains] A, so that
[MATHEMATICAL EXPRESSION OMITTED]
Since this holds for all [e[silon] > 0 we have
μ*(A) ≤ μ*(B).
(iii) Let Ej be a countable collection of subsets of S. For each j choose a countable covering {Aij} of Ej with Aij [member of] Σ and
[MATHEMATICAL EXPRESSION OMITTED]
Since
[MATHEMATICAL EXPRESSION OMITTED]
we have
[MATHEMATICAL EXPRESSION OMITTED]
again, since ε > 0 was arbitrary we have the desired result.
Definition. Let λ be an outer measure on S. A set A is measurable (with respect to λ, or λ-measurable) if
λ(E) = λ(A [intersection] E) + λ(A' [intersection] E)
for all E [subset] S.
It follows that if A is measurable and if E is taken to be S then
λ(S) = λ(A) + λ(A').
The measurability of a set depends on the outer measure function under consideration. There may well be two outer measures for the same space such that a given set A is measurable with respect to one, but not to the other. The following is the important fact we shall need concerning outer measures.
Theorem 4. The measurable subsets of S form a σ-field Σ0, on which λ is a measure.
Proof. [empty set] [member of] Σ0 since λ(E) = λ([empty set]) + λ(E). If A [member of] Σ0, then by the symmetry of the relation
(2) λ(E) = λ(A [intersection] E) + λ(A' [intersection] E)
it trivially follows that A' [member of] Σ0. Now suppose that A and B are in Σ0. Then
(3) [MATHEMATICAL EXPRESSION OMITTED]
Also
[MATHEMATICAL EXPRESSION OMITTED]
and by the measurability of B this is
λ(A [intersection] B' [intersection] E) + λ(A' [intersection] B [intersection] E) + λ(A' [intersection] B' [intersection] E).
This identity and (3) give
λ([A [intersection] B]' [intersection] E) + λ(A [intersection] B [intersection] E) = λ(E).
implying that A [intersection] B [member of] Σ0, and that Σ0 is a field.
Suppose A, B [member of] Σ0, A [intersection] B = [empty set]. If we replace E by ((A [union] B) [intersection] E in (2) we obtain
λ([A [union]] [intersection] E) = λ(A [intersection] E) + λ(B [intersection] E).
By induction, if A1, ..., An [member of] Σ0 and are pairwise disjoint, then for any E [subset] S
(4) [MATHEMATICAL EXPRESSION OMITTED]
Now let {Ai} (i = 1, 2, ...) be a countable collection of members of Σ0 which are pairwise disjoint. For any n
[MATHEMATICAL EXPRESSION OMITTED]
by (4). Since this is true for any n we deduce
[MATHEMATICAL EXPRESSION OMITTED]
and by (iii) this is
[MATHEMATICAL EXPRESSION OMITTED]
We therefore have equality in all of the above statements, implying that [union]∞i=1Ai [member of] Σ0 and that Σ0 is a σ-field. Now substituting
[MATHEMATICAL EXPRESSION OMITTED]
for E we get
[MATHEMATICAL EXPRESSION OMITTED]
so that λ is indeed a measure on Σ0.
Definition. A measure μ on a field Σ of subsets of S is said to be σ-finite if there is a sequence of sets {Ai} [subset] Σ with [union]∞i=1Ai = S and each μ(Ai) < ∞.
Theorem 5 (Hahn Extension Theorem). A measure μ on a field Σ can be extended to a measure on σ(Σ), the σ-field generated by Σ. If μ is σ-finite the extension is unique.
(Continues...)
Excerpted from Lectures on Measure and Integration by Harold Widom. Copyright © 2016 Harold Widom. Excerpted by permission of Dover Publications, Inc..
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