proof of Radon-Nikodym theorem
The following proof of Radon-Nikodym theorem
is based on the original argument by John von Neumann.
We suppose that and are real, nonnegative, and finite.
The extension to the -finite case is a standard exercise,
as is -a.e. uniqueness of Radon-Nikodym derivative.
Having done this, the thesis also holds for signed and complex-valued measures
.
Let be a measurable space
and let two finite measures on
such that for every such that .
Then is a finite measure on
such that if and only if .
Consider the linear functional defined by
(1) |
is well-defined
because is finite and dominated by , so that
it is also linear and bounded because
By Riesz representation theorem
, there exists such that
(2) |
for every .
Then
for every ,
so that - and -a.e.
(Consider the former with or .)
Moreover, the second equality in (LABEL:eq:q)
holds when for ,
thus also when is a simple measurable function
by linearity of integral,
and finally when is a (- and -a.e.)
nonnegative -measurable function
because of the monotone convergence theorem
.
Now, is -measurable and nonnegative - and -a.e.; moreover, - and -a.e. Thus, for every ,
(3) |
Since is finite, , and so is . Then for every
Title | proof of Radon-Nikodym theorem |
---|---|
Canonical name | ProofOfRadonNikodymTheorem |
Date of creation | 2013-03-22 18:58:03 |
Last modified on | 2013-03-22 18:58:03 |
Owner | Ziosilvio (18733) |
Last modified by | Ziosilvio (18733) |
Numerical id | 5 |
Author | Ziosilvio (18733) |
Entry type | Proof |
Classification | msc 28A15 |
Synonym | Hilbert spaces![]() |
Synonym | measure- theoretic proof of Radon-Nikodym theorem |