658 CHAPTER 24. THE BOCHNER INTEGRAL

Thus,

f(∫

Ω

xdµ

)= lim

n→∞f(∫

Ω

yndµ

)= lim

n→∞

∫Ω

f (yn)dµ =∫

Ω

f (x)dµ,

the last equation holding from the dominated convergence theorem (| f (yn)| ≤ ∥ f∥∥yn∥ ≤2∥ f∥∥x∥). This shows 24.6.

It remains to verify 24.8. Let f ∈ X ′. Then from 24.6

f(∫

Ω

(ax(ω)+by(ω))dµ

)=

∫Ω

(a f (x(ω))+b f (y(ω)))dµ

= a∫

Ω

f (x(ω))dµ +b∫

Ω

f (y(ω))dµ

= f(

a∫

Ω

x(ω)dµ +b∫

Ω

y(ω)dµ

).

Since X ′ separates the points of X ,it follows∫Ω

(ax(ω)+by(ω))dµ = a∫

Ω

x(ω)dµ +b∫

Ω

y(ω)dµ

and this proves 24.8. ■A similar result is the following corollary.

Corollary 24.2.6 Let an X valued function x be Bochner integrable. Let L ∈L (X ,Y )where Y is another Banach space. Then Lx is a Y valued Bochner integrable function and

L(∫

Ω

x(ω)dµ

)=∫

Ω

Lx(ω)dµ

Proof: From Theorem 24.2.4 there is a sequence of simple functions {yn} having theproperties listed in that theorem. These are measurable with finitely many values and areforced to be simple because ∥yn∥ ≤ 2∥x∥. Then consider {Lyn} which converges pointwiseto Lx. Since L is continuous and linear,∫

Ω

∥Lyn−Lx∥Y dµ ≤ ∥L∥∫

Ω

∥yn− x∥X dµ

which converges to 0. This implies

limm,n→∞

∫Ω

∥Lyn−Lym∥dµ = 0

and so by definition Lx is Bochner integrable. Also∫Ω

x(ω)dµ = limn→∞

∫Ω

yn (ω)dµ∫Ω

Lx(ω)dµ = limn→∞

∫Ω

Lyn (ω)dµ = limn→∞

L∫

Ω

yn (ω)dµ

Next, ∥∥∥∥L(∫

Ω

x(ω)dµ

)−∫

Ω

Lx(ω)dµ

∥∥∥∥Y

≤∥∥∥∥L(∫

Ω

x(ω)dµ

)−L

∫Ω

yn (ω)dµ

∥∥∥∥Y

+

∥∥∥∥∫Ω

Lyn (ω)dµ−∫

Ω

Lx(ω)dµ

∥∥∥∥Y< ε/2+ ε/2 = ε

whenever n large enough. ■