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Lecture 3: Integration
Integration is a type of “average”.
Definition 1. (a) The integral of a nonnegative simple function ϕ w.r.t. ν is defined as
∫ ϕdν =
∑^ k
i=
aiν(Ai).
(b) Let f be a nonnegative Borel function and let Sf be the collection of all nonnegative simple functions satisfying ϕ(ω) ≤ f (ω) for any ω ∈ Ω. The integral of f w.r.t. ν is defined as (^) ∫
f dν = sup
{∫ ϕdν : ϕ ∈ Sf
} .
(Hence, for any Borel function f ≥ 0, there exists a sequence of simple functions ϕ 1 , ϕ 2 , ... such that 0 ≤ ϕi ≤ f for all i and limn→∞
∫ ϕndν =
∫ f dν.) (c) Let f be a Borel function,
f+(ω) = max{f (ω), 0 }
be the positive part of f , and
f−(ω) = max{−f (ω), 0 }
be the negative part of f. (Note that f+ and f− are nonnegative Borel functions, f (ω) = f+(ω) − f−(ω), and |f (ω)| = f+(ω) + f−(ω).) We say that
∫ f dν exists if and only if at least one of
∫ f+dν and
∫ f−dν is finite, in which case ∫ f dν =
∫ f+dν −
∫ f−dν.
When both
∫ f+dν and
∫ f−dν are finite, we say that f is integrable. Let A be a measurable set and IA be its indicator function. The integral of f over A is defined as ∫
A
f dν =
∫ IAf dν.
A Borel function f is integrable if and only if |f | is integrable.
For convenience, we define the integral of a measurable function f from (Ω, F , ν) to ( R¯, B¯), where R¯ = R ∪ {−∞, ∞}, B¯ = σ(B ∪ {{∞}, {−∞}}). Let A+ = {f = ∞} and A− = {f = −∞}. If ν(A+) = 0, we define
∫ f+dν to be
∫ IAc + f+dν; otherwise
∫ f+dν = ∞.
∫ f−dν is similarly defined. If at least one of
∫ f+dν and
∫ f−dν is finite, then
∫ f dν =
∫ f+dν −
∫ f−dν is well defined.
Notation for integrals∫ f dν =
∫ Ω f dν^ =^
∫ f (ω)dν =
∫ f (ω)dν(ω) =
∫ f (ω)ν(dω). In probability and statistics,
∫ XdP = EX = E(X) and is called the expectation or expected value of X. If F is the c.d.f. of P on (Rk, Bk),
∫ f (x)dP =
∫ f (x)dF (x) =
∫ f dF.
Example 1.5. Let Ω be a countable set, F be all subsets of Ω, and ν be the counting measure For any Borel function f , ∫ f dν =
∑
ω∈Ω
f (ω).
Example 1.6. If Ω = R and ν is the Lebesgue measure, then the Lebesgue integral of f over an interval [a, b] is written as
∫ [a,b] f^ (x)dx^ =^
∫ (^) b a f^ (x)dx, which agrees with the Riemann integral in calculus when the latter is well defined. However, there are functions for which the Lebesgue integrals are defined but not the Riemann integrals.
Properties
Proposition 1.5 (Linearity of integrals). Let (Ω, F , ν) be a measure space and f and g be Borel functions. (i) If
∫ f dν exists and a ∈ R, then
∫ (af )dν exists and is equal to a
∫ f dν. (ii) If both
∫ f dν and
∫ gdν exist and
∫ f dν +
∫ gdν is well defined, then
∫ (f + g)dν exists and is equal to
∫ f dν +
∫ gdν.
A statement holds a.e. ν (or simply a.e.) if it holds for all ω in Nc^ with ν(N) = 0. If ν is a probability, then a.e. may be replaced by a.s.
Proposition 1.6. Let (Ω, F , ν) be a measure space and f and g be Borel. (i) If f ≤ g a.e., then
∫ f dν ≤
∫ gdν, provided that the integrals exist. (ii) If f ≥ 0 a.e. and
∫ f dν = 0, then f = 0 a.e. Proof. (i) Exercise. (ii) Let A = {f > 0 } and An = {f ≥ n−^1 }, n = 1, 2 , .... Then An ⊂ A for any n and limn→∞ An = ∪An = A (why?). By Proposition 1.1(iii), limn→∞ ν(An) = ν(A). Using part (i) and Proposition 1.5, we obtain that
n−^1 ν(An) =
∫ n−^1 IAn dν ≤
∫ f IAn dν ≤
∫ f dν = 0
for any n. Hence ν(A) = 0 and f = 0 a.e.
Consequences: |
∫ f dν| ≤
∫ |f |dν If f ≥ 0 a.e., then
∫ f dν ≥ 0 If f = g a.e., then
∫ f dν =
∫ gdν.
