632 CHAPTER 23. REPRESENTATION THEOREMS

e(N,n,m) where |e(N,n,m)|< ε for all n,m provided N is chosen large enough. This is byuniform integrability which is a consequence of equi-integrability. See Proposition 10.9.6.It follows that∣∣∣∣∫G

gndµ−∫

Ggmdµ

∣∣∣∣ ≤∣∣∣∣∣ N

∑k=1

∫Gk

(gn−gm)dµ

∣∣∣∣∣+ |e(N,n,m)|

<

∣∣∣∣∣ N

∑k=1

∫Gk

(gn−gm)dµ

∣∣∣∣∣+ ε < 2ε

provided n,m are large enough. Thus G is closed with respect to countable disjoint unions.If∫

G gndµ converges, then∫

GC gndµ =∫

Ωgndµ −

∫G gndµ and so

∫GC gndµ converges.

Hence, by Dynkin’s lemma, G ⊇ σ (K ) . For E ∈ σ (K ) define

λ (E) ≡ limn→∞

∫E

gndµ, then λ ≪ µ so there is g such that∫E

gdµ = λ (E) = limn→∞

∫E

gndµ by Radon Nikodym, g ∈ L1

That λ is a measure follows from the above argument that G is closed with respect tocountable disjoint unions.

Now it was just shown that for s a simple function measurable with respect to σ (K ) ,∫sgdµ = lim

n→∞

∫sgndµ.

Can we replace s with h ∈ L∞ (Ω,σ (K ) ,µ)? Letting h be a representative which is uni-formly bounded, there exists a sequence of simple functions {sn} which converges uni-formly to h. ∣∣∣∣∫ hgdµ−

∫hgndµ

∣∣∣∣≤ ∣∣∣∣∫ hgdµ−∫

sgdµ

∣∣∣∣+

∣∣∣∣∫ sgdµ−∫

sgn

∣∣∣∣+ ∣∣∣∣∫ sgndµ−∫

hgndµ

∣∣∣∣The first term on the right is no more than ε ∥g∥L1 because s was chosen to be uniformlywithin ε of h. As to the last term, it is no more than ε maxn ∥gn∥L1 which is no more thanεC since the equi-integrability implies ∥gn∥L1 is bounded. The middle term converges to 0and so limn→∞ |

∫hgdµ−

∫hgndµ|= 0.

Now consider L∞ (Ω,σ (K ) ,µ)i∗← L∞ (Ω,F ,µ)

L1 (Ω,σ (K ) ,µ)i→ L1 (Ω,F ,µ)

where the inclusion map i is

continuous. Let h ∈ L∞ (Ω,F ,µ) so i∗h ∈ L∞ (Ω,σ (K ) ,µ). Then

limn→∞

∫hgndµ = lim

n→∞

∫higndµ = lim

n→∞

∫i∗hgndµ = lim

n→∞

∫i∗hgdµ = lim

n→∞

∫hgdµ

and this shows that gn converges weakly to g. ■One can extend this to an arbitrary measure space by fussing with more details that

involve consideration of a σ algebra which is σ finite.