Page 156 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 12: General Convergence Theorems

Then each f is bounded and Notes
n
f f at each point
n
Since {f } is an increasing sequence of bounded functions such that f f on E
n n
By the monotone convergence theorem

lim f f
n n
E E
For given > 0 a positive integer N such that

f n f for n N
2
E E

f f
N
2
E E
f f
2 N 2
E E

(f f )
N
2
E
Choose
2N

If mA < , then we have

f (f f ) f
= N N
A A

(f f ) f
= N N
A A

(f f ) N as f N
N N
E A

< NmA
2

< N
2

< N
2 2N

=
2 2
=

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