Page 59 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Thus we have proved that g is a measurable function over [a, b] such that
|g| p L [a, b]
Hence g L [a, b]
p
p
Theorem 2: If f L [a, b], p > 1, then f L [a, b]
Proof: f L [a, b] f is measurable over [a, b]
p
Let A = {x [a, b] : |f (x) 1}
1
and A = {x [a, b] : |f (x) 1}
2
Then [a, b] = A A and A A =
1 2 1 2
Using countable additive property of the integrals, we have
b
|f|dx = |f|dx |f|dx … (i)
a A 1 A 2
Now  |f (x)| 1, x A
1
|f| |f| on A as p > 1
p
1
p
p
|f|dx |f| dx as f L [a, b] … (ii)
A 1
A 1
Now |f (x) |<|, x A
2
Using first mean value theorem, we get

|f|dx m (A ) = A finite quantity … (iii)
2
A 2
Combining (ii) and (iii) and making use of (i), we get

b
|f|dx <
a
Thus f is a measurable function over [a, b], such that

b
|f|dx <
a
|f| L [a, b] and hence f L [a, b].
p
Theorem 3: If f L [a, b], g L [a, b]; then f + g L [a, b]
p
p
Proof: Since f, g L [a, b] f, g are measurable over [a, b]
p
f + g is measurable over [a, b]
Let A = {x [a, b] : |f (x)| |g (x)|}
1
and A = {x [a, b] : |f (x)| < |g (x)|}
2
Then [a, b] = A A and A A =
1 2 1 2

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