Page 123 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes At the p stage, we get a measurable set
th
n
A E with m (A ) < and a positive integer N such that
p p–1 p p p
2
1
|f (x) – f (x)| <  n N and x E where
n p p
p
E = E – 1 – A .
p p p

Let A = A p ,
p 1

then m (A) m(A )
p
p 1
n
But m (A ) <
2p
p

n
m (A) < p
2
p 1
1 1
1 a 2 2
But is a G.P. series so S = = 1
2 p 1 r 1 1
p 1 1
2 2
m (A) <

Also, E – A = E – A p
p

= (E A )
p
p

= (E p 1 A )
p
p

= E p
p
Let x E – A. Then x E p and so
p
1
|f (x) – f (x)| < , n N .
n p
p
1
Let us choose p such that < , we get
p

|f (x) – f (x)| < x E – A and n N = N
n p

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