Page 178 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 14: Radon-Nikodym Theorem

Notes
d( 1 2 ) d 1 d 2
d d d
Prove the other result yourself.
Theorem 2: If is a -finite signed measures and is a -finite measure s.t.  , show that

d| | d
.
d d
Proof: Let = – with Hahn decomposition A, B.
+
–
d d d d
Then on A, and on B,
d d d d

d d d d( ) d| |
.
d d d d d

Theorem 3: If be a -finite signed measure and , be -finite measures on (X, A) s.t.  ,
 : then show that

d d d
d d d
Proof: Since we may write = – and
–
+
d d( ) d d( )
, .
d d d d
we need to prove the above result for measures only.

d d
If f and g , (f, g are non-negative functions as obtained in Radon-Nikodym Theorem),
d d
then we need to prove that

(F) = fg d .
F
Let be a measurable simple function s.t.
n
= a i E i ,
i 1

n
then d a i (E i F)
F i 1
n
= a i gd g d .
i 1 F F
E i
Let < > be a sequence of measurable simple function which converges to f, then
n

(F) = fd lim d .
n
F F

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