The Vitali convergence theorem is a convergence theorem which in the case of a finite measure space is
superior to the dominated convergence theorem.
Definition 6.10.2Let (Ω,ℱ,μ) be a measure space and let S ⊆ L^{1}(Ω). S is uniformly integrableif forevery ε > 0 there exists δ > 0 such that for all f ∈S
∫
| fdμ| < ε whenever μ(E) < δ.
E
Lemma 6.10.3If S is uniformly integrable, then |S|≡{|f| : f ∈S} is uniformly integrable. AlsoS is uniformly integrable if S is finite.
Proof:Let ε > 0 be given and suppose S is uniformly integrable. First suppose the functions are real
valued. Let δ be such that if μ
= t^{2}, it follows that if all the functions in S are bounded, then the collection of functions is
uniformly integrable.
The following theorem is Vitali’s convergence theorem.
Theorem 6.10.5Let {f_{n}} be a uniformly integrable set of complex valued functions, μ(Ω) < ∞,andf_{n}(x) → f(x) a.e. where f is a measurable complex valued function. Then f ∈ L^{1}
(Ω)
and
∫
nli→m∞ |fn − f|dμ = 0. (6.10)
Ω
(6.10)
Proof:First it will be shown that f ∈ L^{1}
(Ω )
. By uniform integrability, there exists δ > 0 such that if
μ
(E )
< δ, then
∫
|fn|dμ < 1
E
for all n. By Egoroff’s theorem, there exists a set, E of measure less than δ such that on E^{C},
{fn}
converges uniformly. Therefore, for p large enough, and n > p,
∫
EC |fp − fn|dμ < 1
which implies
∫ ∫
|f |dμ < 1 + |f |dμ.
EC n Ω p
Then since there are only finitely many functions, f_{n} with n ≤ p, there exists a constant, M_{1} such that for
all n,
∫
|fn|dμ < M1.
EC
But also,
∫ ∫ ∫
|fm|dμ = C |fm|dμ + |fm|
Ω ≤ ME + 1 ≡ M. E
1
Therefore, by Fatou’s lemma,
∫ ∫
|f|dμ ≤ lim inf |f |dμ ≤ M,
Ω n→∞ n
showing that f ∈ L^{1} 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}
.
By Egoroff’s theorem, there exists a set, F with μ
(F)
< δ_{1} such that f_{n} converges uniformly to f on
F^{C}. Therefore, there exists N such that if n > N, then
∫
|f − fn|dμ < ε.
FC 3
It follows that for n > N,
∫ ∫ ∫ ∫
|f − f |dμ ≤ |f − f |dμ+ |f|dμ + |f |dμ
Ω n FC n F F n
< ε + ε + ε = ε,
3 3 3