Page 176 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 14: Radon-Nikodym Theorem

Notes
Q
A = A – r .
n o


Evidently, A = A A r ,
r o
where A is disjoint union of measurable sets.

(A) = (A ) + (A ) .
r
r o

Q Q
Obviously A r 1 r
r
n o n o
r r 1
f(x) , x A
n n r
o o
r r 1
(A ) fd (A ) [by first mean value theorem]
n r n r
o o
A r
r r 1
Again (A ) (A ) (A ) , we have
n o r r n o r

1 1
(A ) (A ) fd (A ) (A )
r r r r … (iii)
n n
o o
A r
Now, if (A ) > 0, then g (A ) = 0, [ ( – a ) A is positive, a]
and (A ) = 0 if (A ) = 0 [  )]

In either case, (A ) = f d .
A r
Adding the inequalities (iii) over r, we get

1 1
(A) (A) f d (A) (A ).
r
n n
o o
A
Since n is arbitrary and (A) is assumed to be finite, it follows that
o
(A) f d A  .
A
To show that the theorem is true for -finite measure , decompose X into a countable union of
X of finite -measure. Applying the same argument as above for each X , we get the required
i i
function.
To show the second part, let g be any measurable function satisfying the condition,

(A) = f d A  .
A

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