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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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