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
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 ,
Since is finite, , and so is . Then for every
|Title||proof of Radon-Nikodym theorem|
|Date of creation||2013-03-22 18:58:03|
|Last modified on||2013-03-22 18:58:03|
|Last modified by||Ziosilvio (18733)|
|Synonym||Hilbert spaces proof of Radon-Nikodym’s theorem|
|Synonym||measure- theoretic proof of Radon-Nikodym theorem|