7.10. SOME IMPORTANT GENERAL THEOREMS 177

Proof: First I show S is bounded in L1 (Ω; µ) which means there exists a constant Msuch that for all f ∈S, ∫

Ω

| f |dµ ≤M.

From the properties of h, there exists Rn such that if t ≥ Rn, then h(t)≥ nt. Therefore,∫Ω

| f |dµ =∫[| f |≥Rn]

| f |dµ +∫[| f |<Rn]

| f |dµ

Letting n = 1, ∫Ω

| f |dµ ≤∫[| f |≥R1]

h(| f |)dµ +R1µ ([| f |< R1])

≤ N +R1µ (Ω)≡M.

Next let E be a measurable set. Then for every f ∈S,∫E| f |dµ =

∫[| f |≥Rn]∩E

| f |dµ +∫[| f |<Rn]∩E

| f |dµ

≤ 1n

∫Ω

| f |dµ +Rnµ (E)≤ Nn+Rnµ (E)

and letting n be large enough, this is less than ε/2+Rnµ (E). Now if µ (E) < ε/2Rn, itfollows that for all f ∈S,

∫E | f |dµ < ε . This proves the lemma. ■

Letting h(t)= t2, it follows that if all the functions in S are bounded, then the collectionof functions is uniformly integrable.

The following theorem is Vitali’s convergence theorem.

Theorem 7.10.5 Let { fn} be a uniformly integrable set of complex valued func-tions, µ(Ω)< ∞, and fn(x)→ f (x) a.e. where f is a measurable complex valued function.Then f ∈ L1 (Ω) and

limn→∞

∫Ω

| fn− f |dµ = 0. (7.10)

Proof: First it will be shown that f ∈ L1 (Ω). By uniform integrability, there existsδ > 0 such that if µ (E) < δ , then

∫E | fn|dµ < 1 for all n. By Egoroff’s theorem, there

exists a set, E of measure less than δ such that on EC, { fn} converges uniformly. Therefore,for p large enough, and n > p,

∫EC

∣∣ fp− fn∣∣dµ < 1 which implies∫

EC| fn|dµ < 1+

∫Ω

∣∣ fp∣∣dµ.

Then since there are only finitely many functions, fn with n ≤ p, there exists a constant,M1 such that for all n,

∫EC | fn|dµ < M1. But also,∫

Ω

| fm|dµ =∫

EC| fm|dµ +

∫E| fm| ≤M1 +1≡M.

Therefore, by Fatou’s lemma,∫

Ω| f |dµ ≤ liminfn→∞

∫| fn|dµ ≤ M, showing that f ∈ L1

as hoped.Now S∪{ f} is uniformly integrable so there exists δ 1 > 0 such that if µ (E) < δ 1,

then∫

E |g|dµ < ε/3 for all g ∈ S∪{ f}.