Page 150 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 12: General Convergence Theorems

Notes
= (f f) (f f) as (E – A) A =
n n
E A A

1 f f
2mE n
E A A

m(E A) 2M 1
2mE
A

mE 2M mA as m (E – A) mE
2mE

< 2M
2 4M

=
2 2
=

Thus f n f <
E E
But was arbitrary

lim f = f
n n
E E
12.1.5 Fatou’s Lemma

If {f } is a sequence of non-negative measurable functions and f (x) f (x) almost everywhere on
n n
a set E, then

f lim f
n
n
E E
i.e. f lim inf f
n n
E E
Proof: Since integrals over sets of measure zero are zero.

Without loss of generality, we may assume that the convergence is everywhere. Let h be
a bounded measurable function with h f and h (x) = 0 outside a set E E of finite
measure.
Define a function h by
n
h (x) = Min. {h(x), f (x)}
n n
then h (x) h(x) and h (x) f (x)
n n n
h is bounded by the boundedness of h and vanishes outside E as x E – E h(x) = 0
n
h (x) = 0 because
n

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