Page 153 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
= 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
n
E E

f lim f n … (1)
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

From (1) and (2), we have

f lim f lim f f
n n
n n
f lim f
n n

Theorem 2: Let {u } be a sequence of non-negative measurable functions, and let f = u .
n n
h 1
Then f u n
h 1
n
Proof: Let f = u + u + … + u = u
n 1 2 n j
j 1
then f f
n
i.e. lim f f
n n
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

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