Page 174 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 14: Radon-Nikodym Theorem

Notes

Notes

 If is -finite, the converse is also true.
 If and are signed measures on (X, ), then  if | |  | |.

Radon-Nikodym Theorem

Let (X, , ) be a -finite measure space. If be a measure defined on A s.t. is absolutely
continuous w.r.t. , then there exists a non-negative measurable function f on s.t.

(A) = f d , A  .
A
The function f is unique in the sense that if g is any measurable function with the property
defined as above, then f = g almost everywhere with respect to .
Proof: To establish the existence of the function f, we shall use the following two Lemmas:
Lemma 1: Let E be a countable set of real numbers. Let for each a E there is a set F  s.t.
a
F F , whenever b < a i.e. <F > is a monotonic decreasing sequence of subsets of  corresponding
a b n
to the sequence <a > of real numbers in E. Then a measurable extended real valued function f
n
on X s.t.
f (x) a, x F ,
a
and f (x) a, x (X – F ).
a
Proof: Let f (x) = inf {a : x F } x X and let, conventionally
a
inf {empty collection of real numbers} =
Now, x F f(x) a
a
x F x F for every b < a
a a
f (x) a
Now, f (x) < a x F for some b < a
b

or {x : f (x) < a} = [F ]
b .
b a
Also x F f (x) b < a for some b < a.
b
Hence f is measurable.
Again, by definition of f, we observe that

f (x) a, x F ,
a
and f (x) a, x F .
a
Thus f is the required function.
Lemma 2: Let E be a countable set of real numbers. Let corresponding to each a E, there is a set
F  s.t.
a
(F – F ) = 0 whenever b > a.
a b
Then there exists a measurable function f with the property
x F f (x) a a.e.
a
and x (X – F ) f (x) > a a.e.
a

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