256 CHAPTER 11. ABSTRACT MEASURE AND INTEGRATION

If L1(E) is written, the σ algebra is defined as

{E ∩A : A ∈F}

and the measure is µ restricted to this smaller σ algebra. Clearly, if f ∈ L1(Ω), then

f XE ∈ L1(E)

and if f ∈ L1(E), then letting f̃ be the 0 extension of f off of E, it follows f̃ ∈ L1(Ω).

11.5 Vitali Convergence TheoremThe Vitali convergence theorem is a convergence theorem which in the case of a finitemeasure space is superior to the dominated convergence theorem.

Definition 11.5.1 Let (Ω,F ,µ) be a measure space and let S ⊆ L1(Ω). S is uniformlyintegrable if for every ε > 0 there exists δ > 0 such that for all f ∈S

|∫

Ef dµ|< ε whenever µ(E)< δ .

Lemma 11.5.2 If S is uniformly integrable, then |S| ≡ {| f | : f ∈ S} is uniformly inte-grable. Also S is uniformly integrable if S is finite.

Proof: Let ε > 0 be given and suppose S is uniformly integrable. First suppose thefunctions are real valued. Let δ be such that if µ (E)< δ , then∣∣∣∣∫E

f dµ

∣∣∣∣< ε

2

for all f ∈S. Let µ (E)< δ . Then if f ∈S,∫E| f |dµ ≤

∫E∩[ f≤0]

(− f )dµ +∫

E∩[ f>0]f dµ

=

∣∣∣∣∫E∩[ f≤0]f dµ

∣∣∣∣+ ∣∣∣∣∫E∩[ f>0]f dµ

∣∣∣∣<

ε

2+

ε

2= ε.

In general, if S is a uniformly integrable set of complex valued functions, the inequalities,∣∣∣∣∫ERe f dµ

∣∣∣∣≤ ∣∣∣∣∫Ef dµ

∣∣∣∣ , ∣∣∣∣∫EIm f dµ

∣∣∣∣≤ ∣∣∣∣∫Ef dµ

∣∣∣∣ ,imply ReS ≡ {Re f : f ∈S} and ImS ≡ {Im f : f ∈S} are also uniformly integrable.Therefore, applying the above result for real valued functions to these sets of functions, itfollows |S| is uniformly integrable also.