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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 \mu and \nu are two finite measures on (\Omega,\mathcal{F}) then there exists a non-negative (w.r.t. both measures) measurable function f and a \mu-null set B such that

\nu(A)=\int_A f \, d\mu+ \nu(A \cap B)

for each A \in \mathcal{F}.

Proof:

Let \pi:=\mu+\nu and consider the operator

T(f):=\int f\, d\nu. Read the rest of this entry »

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