9.9. SOME IMPORTANT GENERAL THEORY 215

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.

For the last part, is suffices to verify a single function in L1 (Ω) is uniformly integrable.To do so, note that from the dominated convergence theorem, limR→∞

∫[| f |>R] | f |dµ = 0.

Let ε > 0 be given and choose R large enough that∫[| f |>R] | f |dµ < ε

2 . Now let µ (E)< ε

2R .Then ∫

E| f |dµ =

∫E∩[| f |≤R]

| f |dµ +∫

E∩[| f |>R]| f |dµ

< Rµ (E)+ε

2<

ε

2+

ε

2= ε.

This proves the lemma. ■The following gives a nice way to identify a uniformly integrable set of functions.

Lemma 9.9.4 Let S be a subset of L1 (Ω,µ) where µ (Ω) < ∞. Let t → h(t) be acontinuous function which satisfies limt→∞

h(t)t = ∞. Then S is uniformly integrable and

bounded in L1 (Ω) if sup{∫

Ωh(| f |)dµ : f ∈S}= N < ∞.

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. Lettingn = 1, and f ∈S,∫

Ω

| f |dµ =∫[| f |≥R1]

| f |dµ +∫[| f |<R1]

| f |dµ

≤∫[| f |≥R1]

h(| f |)dµ +R1µ ([| f |< R1])≤ N +R1µ (Ω)≡M. (9.10)

Next let E be a measurable set. Then for every f ∈S, it follows from 9.10∫E| f |dµ =

∫[| f |≥Rn]∩E

| f |dµ +∫[| f |<Rn]∩E

| f |dµ

≤ 1n

∫Ω

| f |dµ +Rnµ (E)≤ Mn+Rnµ (E) (9.11)

Let n be large enough that M/n < ε/2 and then let µ (E) < ε/2Rn. Then 9.11 is less thanε/2+Rn (ε/2Rn) = ε ■

Letting h(t)= t2, it follows that if all the functions in S are bounded, then the collectionof functions is uniformly integrable. Another way to discuss uniform integrability is thefollowing. This other way involving equi-integrability is used a lot in probability.

Definition 9.9.5 Let (Ω,F ,µ) be a measure space with µ (Ω) < ∞. A set S ⊆L1 (Ω) is said to be equi-integrable if for every ε > 0 there exists λ > 0 sufficiently large,such that

∫[| f |>λ ] | f |dµ < ε for all f ∈S.