Kenneth Kuttler

660 CHAPTER 21. THE BOCHNER INTEGRAL

Lemma 21.3.3 The above definition is well defined. Furthermore, if 21.3.12 holds thens→∥A(s)∥ is measurable and if 21.3.13 holds, then∥∥∥∥∫

Ω

A(s)dµ

∥∥∥∥≤ ∫Ω

∥A(s)∥dµ.

Proof: It is clear that in case s→ A(s)x is measurable for all x∈ X there exists a uniqueΨ ∈L (X ,Y ) such that

Ψ(x) =∫

Ω

A(s)xdµ.

This is because x→∫

ΩA(s)xdµ is linear and continuous. It is continuous because∥∥∥∥∫

Ω

A(s)xdµ

∥∥∥∥≤ ∫Ω

∥A(s)x∥dµ ≤∫

Ω

∥A(s)∥dµ ∥x∥

Thus Ψ =∫

ΩA(s)dµ and the definition is well defined.

Now consider the assertion about s→∥A(s)∥. Let D′ ⊆ B′ the closed unit ball in Y ′ besuch that D′ is countable and

∥y∥= supy∗∈D′

|y∗ (y)| .

This is from Lemma 21.1.6. Recall X is separable. Also let D be a countable dense subsetof B, the unit ball of X . Then

{s : ∥A(s)∥> α} =

{s : sup

x∈D∥A(s)x∥> α

}= ∪x∈D {s : ∥A(s)x∥> α}= ∪x∈D

(∪y∗∈D′ {|y∗ (A(s)x)|> α}

)and this is measurable because s→ A(s)x is strongly, hence weakly measurable.

Now suppose 21.3.13 holds. Then for all x,∫Ω

∥A(s)x∥dµ <C∥x∥ .

It follows that for ∥x∥ ≤ 1,∥∥∥∥(∫Ω

A(s)dµ

)(x)∥∥∥∥= ∥∥∥∥∫

Ω

A(s)xdµ

∥∥∥∥≤ ∫Ω

∥A(s)x∥dµ ≤∫

Ω

∥A(s)∥dµ

and so ∥∥∥∥∫Ω

A(s)dµ

∥∥∥∥≤ ∫Ω

∥A(s)∥dµ.

Now it is interesting to consider the case where A(s) ∈L (H,H) where s→ A(s)x isstrongly measurable and A(s) is compact and self adjoint. Recall the Kuratowski measur-able selection theorem, Theorem 11.1.11 on Page 228 listed here for convenience.

660 CHAPTER 21. THE BOCHNER INTEGRALLemma 21.3.3 The above definition is well defined. Furthermore, if 21.3.12 holds thengs —> ||A (s)|| is measurable and if 21.3.13 holds, then[acer [coteProof: It is clear that in case s — A (s)x is measurable for all x € X there exists a uniqueWe L (X,Y) such thatY (x) = [ As)xau.This is because x + fA (s)xdw is linear and continuous. It is continuous because| [acoan| < [lacoalian sf lacsyian iThus ¥ = /,A(s)d and the definition is well defined.Now consider the assertion about s — ||A (s)|]. Let D’ C B’ the closed unit ball in Y’ besuch that D’ is countable andIly|| = sup |y*(y)]-y*eD!This is from Lemma 21.1.6. Recall X is separable. Also let D be a countable dense subsetof B, the unit ball of X. Then{s:|A(s)| >a} = {sup > aUxep {s: ||A (s) x|| > a}= Uren (Uren {Iy* (A(s)x)| > })and this is measurable because s — A (s)x is strongly, hence weakly measurable.Now suppose 21.3.13 holds. Then for all x,[la@alian <clal.It follows that for ||x|| < 1,(faa) o_ | [ A(s)xa < [lAG)alan< [ lA()lianand so| Lacan < [lA @)|idu. rNow it is interesting to consider the case where A(s) € &(H,H) where s + A(s)x isstrongly measurable and A (s) is compact and self adjoint. Recall the Kuratowski measur-able selection theorem, Theorem 11.1.11 on Page 228 listed here for convenience.