CHAPTER 2

Measurable Functions

In developing the Lebesgue integral we shall be concerned with classes of real-valued functions defined on a set X. In various applications the set X may be the unit interval I = [0, 1] consisting of all real numbers X satisfying 0 X 1; it may be the set N = {1, 2, 3,…} of natural numbers; it may be the entire real line R; it may be all of the plane; or it may be some other set. Since the development of the integral does not depend on the character of the underlying space X, we shall make no assumptions about its specific nature.

Given the set X, we single out a family X of subsets of X which are “well-behaved” in a certain technical sense. To be precise, we shall assume that this family contains the empty set ∅ and the entire set X, and that X is closed under complementation and countable unions.

2.1 DEFINITION. A family X of subsets of a set X is said to be a σ-algebra (or a σ-field) in case:

(i) ∅, X belong to X. (ii) If A belongs to X, then the complement ( A) = X \ A belongs to X. (iii) If ( An) is a sequence of sets in X, then the union belongs to X.

An ordered pair ( X, X) consisting of a set X and a σ-algebra X of subsets of X is called a measurable space. Any set in X is called an X-measurable set, but when the σ-algebra X is fixed (as is generally the case), the set will usually be said to be measurable.

The reader will recall the rules of De Morgan:

(2.1)

It follows from these that the intersection of a sequence of sets in X also belongs to X.

We shall now give some examples of σ-algebras of subsets.

2.2 EXAMPLES, (a) Let X be any set and let X be the family of all subsets of X.

(b) Let X be the family consisting of precisely two subsets of X, namely ∅ and X. (c) Let X = {1, 2, 3,…} be the set N of natural numbers and let X consist of the subsets (d) Let X be an uncountable set and X be the collection of subsets which are either countable or have countable complements. (e) If X1 and X2 are a-algebras of subsets of X, let X3 be the intersection of X1 and X2; that is, X3 consists of all subsets of X which belong to both X1 and X2. It is readily checked that X3 is a a-algebra. (f) Let A be a nonempty collection of subsets of X. We observe that there is a smallest a-algebra of subsets of X containing A. To see this, observe that the family of all subsets of A is a σ-algebra containing A and the intersection of all the σ-algebras containing A is also a σ-algebra containing A. This smallest σ-algebra is sometimes called the σ-algebra generated by A. (g) Let X be the set R of real numbers. The Borel algebra is the σ-algebra B generated by all open intervals ( a, b) in R. Observe that the Borel algebra B is also the σ-algebra generated by all closed intervals [ a, b] in R. Any set in B is called a Borel set. (h) Let X be the set of extended real numbers. If E is a Borel subset of R, let

(2.2)

and let be the collection of all sets E, E1 , E2, E3 as E varies over B. It is readily seen that is a σ-algebra and it will be called the extended Borel algebra.

In the following, we shall consider a fixed measurable space ( X, X).

2.3 DEFINITION. A function f on X to R is said to be X-measurable (or simply measurable) if for every real number α the set

(2.3)

belongs to X.

The next lemma shows that we could have modified the form of the sets in defining measurability.

2.4 Lemma. The following statements are equivalent for a function f onX to R:

(a) For every α ∈ R, the set Aα = { x ∈ X: f( x) > α} belongs to X. (b) For every α ∈ R, the set Bα = { x ∈ X: f( x) α} belongs to X. (c) For every α ∈ R, the set Cα = { x ∈ X: f( x) ≥ ; α} belongs to X. (d) For every α ∈ R, the set Dα = { x ∈ X: f( x) < α} belongs to X.

PROOF. Since Bα and A α are complements of each other, statement (a) is equivalent to statement (b). Similarly, statements (c) and (d) are equivalent. If (a) holds, then Aα – 1/n belongs to X for each N and since

it follows that Cα ∈ X. Hence (a) implies (c). Since

it follows that (c) implies (a).

2.5 Examples, (a) Any constant function is measurable. For, if/(jc) = c for all X ∈ X and if α ≥ c, then

whereas if α < c, then

(b) If E ∈ X, then the characteristic function X E, defined by

is measurable. In fact, { x ∈ X: Xe( x) > α}is either X, E, or ∅.

(c) If X is the set R of real numbers, and X is the Borel algebra B, then any continuous function f on R to R is Borel measurable (that is, B-measurable). In fact, if f is continuous, then { x ∈ R: f( x) > α} is an open set in R and hence is the union of a sequence of open intervals. Therefore, it belongs to B. (d) If X = R and X = B, then any monotone function is Borel measurable. For, suppose that f is monotone increasing in the sense that X x’ implies f( x) f( x’). Then { x ∈ R: f( x) > α} consists of a half-line which is either of the form { x ∈ R: X > α} or the form { x ∈ R: X ≥ a}, or is R or ∅.

Certain simple algebraic combinations of measurable functions are measurable, as we shall now show.

2.6 LEMMA. Let f and g be measurable real-valued functions and let c be a real number. Then the functions

are also measurable.

PROOF, (a) If c = 0, the statement is trivial. If c > 0, then

The case c < 0 is handled similarly.

(b) If α < 0, then { x ∈ X: ( f( x))2 > α} = X; if α ≥ 0, then (c) By hypothesis, if R is a rational number, then

belongs to X. Since it is readily seen that

it follows that f + g is measurable.

(d) Since , it follows from parts (a), (b), and (c) that f g is measurable. (e) If α < 0, then { x ∈ X: |f( x)| > α} = X, whereas if α ≥ 0, then

Thus the function | f| is measurable.

If f is any function on X to R, let f + and f − be the nonnegative functions defined on X by

(2.4)

The function f + is called the positive part of f and f − is called the negative part of f. It is clear that

(2.5)

and it follows...