Page 154 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 12: General Convergence Theorems

then h (x) h(x) and h (x) f (x) Notes
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
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 measurable function 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

= lim f
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 lim f n
h f n … (1)
E E

f lim f n
n
E E
Since {f } is monotonically increasing sequence and f f
n n
f f
n
f f
n

lim f f … (2)
n n

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