Page 177 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes For each n N, define
1
A = x X : f(x) g(x) 
n
n
1
and B = x X :g(x) f(x)  .
n n

1
Since f (x) – g (x) ,  x A n , we have by first mean value theorem
n
1
(f g)d (A )
n
n
A n
1
fd gd (A )
n
n
A n A n
1 1
(A ) – (A ) (A ) or 0 (A )
n n n n n n
(A ) 0.
n
Since (A ) is always greater than equal to zero, we have (A ) = 0.
n n
Similarly, we can show that
(B ) 0.
n
If C = {x X : f (x) g (x)}

=  (A  B ),
n
n
n 1
then (C) = 0 f = g a.e. on X w.r.t. .
Hence the theorem.
Theorem 1: If , are -finite signed measures on (X, A) and  , , then
1 2 1 2
d( ) d d d d( )
1 2 1 2 and 1 1
d d d d d
Proof: Since , are -finite and  ,  , we have that + is also -finite and
1 2 1 2 1 2
+  .
1 2
Now for any A ,
( + ) (A) = (A) + (A)
1 2 1 2

d d d d
= 1 d 2 d 1 2 d
d d d d
A A A
d d d
1 2 d 1 2 d
d d d
A A

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