Page 151 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Since h = h or h = f
n n n
h is measurable function on E
n
If h = h, then h h
n n
If h = f < h f
n n
then f h as f f
n n
h h
n
Thus h h
n
Since h (x) h (x) for each x E and {h } is a sequence of bounded measurable functions
n n
on E
By Bounded Convergence Theorem

h h h h lim h
n n
E E E E E E
as E = (E – E ) E & (E – E ) E =

lim h
= n
n
E
lim f as h f
n n n
n
E
lim f as E E
n
n
E
h lim f
n
n
E E
Taking supremum over all h f, we get

sup h h lim f n
n f n
E E E

f f lim f n
n
E E E
Remarks:
(1) If in Fatou’s Lemma, we take

1, n x n 1
f (x) =
n 0, otherwise
with E = R

then f lim f n
n
E E
Thus in Fatou’s Lemma, strict inequality is possible.

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