Page 82 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 7: Convergence and Completeness

Now we define a function f such that Notes
f (x) = 0, x E

and f (x) = g (x) + g g , for x [a, b] but x E,
1 k 1 k
k 1
m 1
or f (x) = lim g 1 g g k , for x E
m k 1
k 1
= lim g (x)
m m
Thus f (x) = 0, for x E and

f (x) = lim g (x) for x E.
m m
f (x) = lim g (x) a.e. in [a, b]
m
m
or lim g m f = 0 a.e. in [a, b] … (iii)
m
m 1
Also, g (x) = g g g
m 1 k 1 k
k 1
m 1
|g | |g | + g g
m 1 k 1 k
k 1

|g | + g g k = g,
1 k 1
k 1
|g | g, m N
m
lim g (x) g
m
m
(iii) |f| g.
Again, |g – f| |g | + |f| g + g = 2g.
m m
|g – f| 2 g.
m
p
Thus there exists a function g L [a, b] s.t.
|g – f| 2g, m
m
and lim g m f = 0 a.e. in [a, b] … (iv)
m
Applying Lebesgue dominated convergence theorem,

b b b
p
p
lim g m f dx = lim g m f dx 0 dx 0 [Using (iv)]
m m
a a a

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