Page 152 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 12: General Convergence Theorems

(2) f 0 x E is essential for Fatou’s Lemma Notes
n
However, if we take

1 2
n, x
f (x) = n n
n
0, otherwise
with E = [0, 2]

Then f lim f .
n
n
E E
12.1.6 Monotone Convergence Theorem

Statement: Let {f } be an increasing sequence of non-negative measurable functions and let
n
f = lim f . Then
n n

f lim f
n n
Proof: 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
n
x E – E h(x) = 0 h (x) = 0 because f (x) 0
n n
Since h = h or h = f
n n n
h is measurable function on E
n
If h = h, then h h
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 measurable functions on E
n n
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

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