656 CHAPTER 24. THE BOCHNER INTEGRAL

If x is Bochner integrable, define∫Ω

x(ω)dµ ≡ limn→∞

∫Ω

xn (ω)dµ. (24.5)

First it is important to show that this integral is well defined. When this is done, aneasier to use condition will be developed. Note that by Lemma 24.1.2, if x is stronglymeasurable, ∥x∥ is a measurable real valued function. Thus, it makes sense to consider∫

Ω∥x∥dµ and also

∫Ω∥x− xn∥dµ .

Theorem 24.2.4 The definition of Bochner integrability is well defined. Also, astrongly measurable function x is Bochner integrable if and only if

∫Ω∥x∥dµ < ∞. In this

case that the function is Bochner integrable, an approximating sequence of simple functions{yn} exists such that ∥yn (ω)∥ ≤ 2∥x(ω)∥ for all ω and

limn→∞

∫Ω

∥yn (ω)− x(ω)∥dµ = 0

Proof:⇒First consider the claim about the integral being well defined. Let {xn} be asequence of simple functions converging pointwise to x and satisfying the conditions givenabove for x to be Bochner integrable. Then∣∣∣∣∫

Ω

∥xn (ω)∥dµ−∫

Ω

∥xm (ω)∥dµ

∣∣∣∣≤ ∫Ω

∥xn− xm∥dµ

which is given to converge to 0 as n,m→ ∞ which shows that {∫

Ω∥xn (ω)∥dµ}∞

n=1 is aCauchy sequence. Hence it is bounded and so, by Fatou’s lemma,∫

Ω

∥x(ω)∥dµ ≤ lim infn→∞

∫Ω

∥xn (ω)∥dµ < ∞

The limit in 24.5 exists because∥∥∥∥∫Ω

xndµ−∫

Ω

xmdµ

∥∥∥∥= ∥∥∥∥∫Ω

(xn− xm)dµ

∥∥∥∥≤ ∫Ω

∥xn− xm∥dµ

and the last term is no more than ε whenever n,m are large enough. From Fatou’s lemma,if n is large enough, ∫

Ω

∥xn− x∥dµ < ε

Now if you have another sequence {x̂n} satisfying the condition 24.4 along with point-wise convergence to x,∥∥∥∥∫

Ω

xndµ−∫

Ω

x̂ndµ

∥∥∥∥ =

∥∥∥∥∫Ω

(xn− x̂n)dµ

∥∥∥∥≤ ∫Ω

∥xn− x̂n∥dµ

≤∫

Ω

∥xn− x∥dµ +∫

Ω

∥x− x̂n∥dµ < 2ε

if n is large enough. Hence convergence of the integrals of the simple functions takes placeand these integrals converge to the same thing. Thus the definition is well defined and∫

Ω∥x∥dµ < ∞.⇐Next suppose

∫Ω∥x∥dµ < ∞ for x strongly measurable. By Lemma 24.1.2, there

is a sequence of finite valued measurable functions {yn} with ∥yn (ω)∥ ≤ 2∥x(ω)∥ and