The following proof of Radon-Nikodym theorem is based on the original argument by John von Neumann. We suppose that JG-0-JG and JG-1-JG are real, nonnegative, and finite. The extension to the JG-2-JG -finite case is a standard exercise, as is JG-3-JG -a.e. uniqueness of Radon-Nikodym derivative. Having done this, the thesis also holds for signed and complex-valued measures.

Let JG-4-JG be a measurable space and let JG-5-JG two finite measures on JG-6-JG such that JG-7-JG for every JG-8-JG such that JG-9-JG . Then JG-10-JG is a finite measure on JG-11-JG such that JG-12-JG if and only if JG-13-JG .

Consider the linear functional defined by JG-14-JGJG-15-JG is well-defined because JG-16-JG is finite and dominated by JG-17-JG , so that it is also linear and bounded because By Riesz representation theorem, there exists JG-18-JG such that JG-19-JGfor every JG-20-JG . Then for every JG-21-JG , so that JG-22-JG JG-23-JG - and JG-24-JG -a.e. (Consider the former with JG-25-JG or JG-26-JG .) Moreover, the second equality in () holds when JG-27-JG for JG-28-JG , thus also when JG-29-JG is a simple measurable function by linearity of integral, and finally when JG-30-JG is a (JG-31-JG - and JG-32-JG -a.e.) nonnegative JG-33-JG -measurable function because of the monotone convergence theorem.

Now, JG-34-JG is JG-35-JG -measurable and nonnegative JG-36-JG - and JG-37-JG -a.e.; moreover, JG-38-JG JG-39-JG - and JG-40-JG -a.e. Thus, for every JG-41-JG , JG-42-JGSince JG-43-JG is finite, JG-44-JG , and so is JG-45-JG . Then for every JG-46-JG