The Real And Complex Number Systems

which implies that
∫ b m∑
a
k=1
|f nk − g| 2 dx <
which implies that, by Fatou’s lemma,
m∑
k=1
1
2 k
∫ b
a
lim inf ∑ m ∫ b m∑
|f nk − g| 2 dx ≤ lim inf |f nk − g| 2 dx
m→∞ m→∞
k=1
a
k=1
∞∑
∫ b
= |f nk − g| 2 dx < 1.
k=1
a
That is,
which implies that
∫ b ∞∑
a
k=1
|f nk − g| 2 dx < 1
∞∑
|f nk − g| 2 < ∞ a.e. on [a, b]
k=1
which implies that f nk → g a.e. on [a, b] .
Note: The reader can see the book, Measure and Integral (An Introduction
to Real Analysis) written by Richard L. Wheeden and
Antoni Zygmund, pp 75.
(4) There is another proof by using Egorov’s Theorem: Let {f k } be a
measurable functions defined on a finite measurable set E with finite limit
function f. Then given ε > 0, there exists a closed set F (⊆ E) , where
|E − F | < ε such that
f k → f uniformly on F.
Proof: If f ≠ g on [a, b] , then h := |f − g| ≠ 0 on [a, b] . By continuity
of h, there exists a compact subinterval [c, d] such that |f − g| ≠ 0. So, there
exists m > 0 such that h = |f − g| ≥ m > 0 on [c, d] . Since
∫ b
a
|f n − g| 2 dx → 0 as n → ∞,
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