Page 81 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 1
In particular, m > n f – f < k .
k m n k 2
Obviously n < n < n … < n < …
1 2 3 k
i.e. <n > is a monotonic increasing sequence of natural numbers.
k
Set g = f , then from above, we have
k n k
1
g – g = f f ,
2 1 p n 2 n 2 p 2
1
g – g = f f ,
3 2 p n 3 n 2 p 2 2
… … … … …
… … … … …

1
g – g = f f .
k + 1 k p n k 1 n k p k
2
… … … … …
… … … … …

Adding these inequalities, we get

1
g k 1 g k p < k 1 … (i)
k 1 k 1 2

Thus g g is convergent. Define g such that
k 1 k p
k 1

g (x) = g (x) g k 1 g k if R.H.S. is convergent … (ii)
1
k 1 p
and g (x) = , if right hand side is divergent.
1
b b n p
p
Now, |g(x)| dx = lim |g (x) g g p
n 1 k 1 k dx
a a k 1
n
or g = lim g g g (By Minkowski’s inequality)
p 1 p k 1 k
n p
k 1

= g + g k 1 g k p < g + 1, [by (i)]
1 p 1 p
k 1
p
g < g L [a, b].
p
Let E = {x [a, b] : g (x) = }.

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