Page 161 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Solution: From the given hypothesis it follows that

lim f f … (1)
n
Also by Fatou’s Lemma, we have

f lim f n … (2)

Then from (1) and (2), we get

f lim f n lim f n f .

Hence f = lim f n lim f n lim f n .

n n
x
x
1
1
Example: If > 0, prove that Lim 1 x dx e .x dx , where the integrals are
n n
o o
taken in the Lebesgue sense.
x n x n x
–x
Solution: If f (x) = 1 .x 1 0 , then f (x) g(x), where g(x) = e .x recall Lim 1 e
–1
n n n n
n
Also g(x) L[0, ], hence by Lebesgue dominated convergence theorem, we get
n
Lim f (x) dx = Lim f (x) dx
n
n n n
o o
n
x 1
= Lim 1 .x dx
n n
o
x
= e .x 1 dx
o

Example: Show that if > 1,

1
xsin x 1
dx 0(n ) as n .
1 (nx)
o
Solution: Consider the sequence <f (x)> s.t.
n
nx sin x
f (x) = , n = 1, 2, ………
n 1 (nx)
Obviously since > 1, and x [0, 1]

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