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There is a remarkably nice proof of the Lebesgue decomposition theorem (described below) by von Neumann. This leads immediately to the Radon-Nikodym theorem.

**Theorem:**

*If and are two finite measures on then there exists a non-negative (w.r.t. both measures) measurable function and a -null set such that*

*for each .*

**Proof:**

Let and consider the operator

.* Read the rest of this entry »*

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