Kenneth Kuttler

652 CHAPTER 21. THE BOCHNER INTEGRAL

Since f ∈ X ′ is arbitrary, and X ′ separates the points of X , it follows that ∑nk=1 akµ (Ek) = 0

as hoped. It is now obvious that the integral is linear on simple functions.As to the triangle inequality, say x(s) = ∑

nk=1 akXEk (s) . Then from the triangle in-

equality, ∥∥∥∥∫Ω

x(s)dµ

∥∥∥∥=∥∥∥∥∥ n

∑k=1

akµ (Ek)

∥∥∥∥∥≤ n

∑k=1∥ak∥µ (Ek) =

∫Ω

∥x(s)∥dµ

Definition 21.2.3 A strongly measurable function x is Bochner integrable if there exists asequence of simple functions xn converging to x pointwise and satisfying∫

Ω

∥xn (s)− xm (s)∥dµ → 0 as m,n→ ∞. (21.2.3)

If x is Bochner integrable, define∫Ω

x(s)dµ ≡ limn→∞

∫Ω

xn (s)dµ. (21.2.4)

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 21.1.2, 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 21.2.4 The definition of Bochner integrability is well defined. Also, a stronglymeasurable function x is Bochner integrable if and only if

∫Ω∥x∥dµ < ∞. In this case

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

limn→∞

∫Ω

∥yn (s)− x(s)∥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 (s)∥dµ−∫

Ω

∥xm (s)∥dµ

∣∣∣∣≤ ∫Ω

∥xn− xm∥dµ

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

Ω∥xn (s)∥dµ}∞

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

Ω

∥x(s)∥dµ ≤ lim infn→∞

∫Ω

∥xn (s)∥dµ < ∞

The limit in 21.2.4 exists because∥∥∥∥∫Ω

xndµ−∫

Ω

xmdµ

∥∥∥∥= ∥∥∥∥∫Ω

(xn− xm)dµ

∥∥∥∥≤ ∫Ω

∥xn− xm∥dµ

652 CHAPTER 21. THE BOCHNER INTEGRALSince f € X’ is arbitrary, and X’ separates the points of X, it follows that )2_, ax (Ex) = 0as hoped. It is now obvious that the integral is linear on simple functions.As to the triangle inequality, say x(s) = )%_, a 2x, (s). Then from the triangle in-equality,\/ v(s)dal =||)° acu (Ex)2 k=l<¥ lla E,) = [ |lx(s)|ld< Yi lalla a) = fi lie(oldnDefinition 21.2.3 A strongly measurable function x is Bochner integrable if there exists asequence of simple functions X, converging to x pointwise and satisfying| \Xn (8) —Xm (s)|| dL > 0 as m,n > ©. (21.2.3)QIf x is Bochner integrable, define[x@an= lim [| x,(s)du. (21.2.4)Q ne JOFirst 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 21.1.2, if x is stronglymeasurable, ||x|| is a measurable real valued function. Thus, it makes sense to consider[ollx\|dp and also J || —an|| dU.Theorem 21.2.4 The definition of Bochner integrability is well defined. Also, a stronglymeasurable function x is Bochner integrable if and only if Ja ||x||du < °. In this casethat the function is Bochner integrable, an approximating sequence {y,} exists such thatlyn (5)|| <2|kx(s)]| for all s andtim [ ym (s) —x(9)||du =0n—ooProof: =First consider the claim about the integral being well defined. Let {x,} be asequence of simple functions converging pointwise to x and satisfying the conditions givenabove for x to be Bochner integrable. Then[ls corlan— fin (sllas| < fn alldQ Q Qwhich is given to converge to 0 as n,m — c which shows that { fo ||xn (s)||du}y_, is aCauchy sequence. Hence it is bounded and so, by Fatou’s lemma,[i lxolldu <tim int, lbsn(s) lau <=Q neo JOThe limit in 21.2.4 exists because| [van — [sna = l/ (4 —in) a < | \|Xn —Xm|| duQ Q Q Q