Page 157 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
f
A

12.1.7 Lebesgue Dominated Convergence Theorem

Theorem 4: State and prove Lebesgue dominated convergence theorem

Statement: Let g be an integrable function on E and let {f } be a sequence of measurable functions
n
such that |f | g on E and lim f = f a.e. on E. Then
n n n
f lim f
n n .
E E
Proof: Since we know that if f is a measurable function over a set E and there is an integrable
function g such that |f| g, then f is integrable over E. So clearly, each f is integrable over E.
n
Also lim f = f a.e. on E.
n n
and |f | g a.e. on E
n
|f| g a.e. on E.

Hence f is integrable over E.
Let { } be a sequence of functions defined by = f . Clearly, is a non-negative and integrable
n n n + g n
function for each n.
Therefore, by Fatou’s Lemma, we have

(f g) lim (f n g)
n
E E

f lim f n … (1)
n
E E
Similarly, let { } be a sequence of functions defined by = g – f . Clearly is a non-negative
n n n n
and integrable function for each n. So, given by Fatou’s Lemma, we have
(g f) lim (g f )
n
n
E E
g f g lim f n
n
E E E E

f lim f
n
n
E E
f lim f … (2)
n
E E
Hence from (1) & (2), we get

f = lim f n lim f n
E E E

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