Radon-Nikodym theorem

The Radon-Nikodym theorem states that given two measures $𝜇, 𝜈$ on a measurable space $(𝑋, Σ)$ such that ^eq, there exists a measurable function $𝑓 : 𝑋 \to [0, \infty)$ such that for any $𝐴 \in Σ$ measure

𝜈 (𝐴) = \int 𝐴 𝑓 (𝑥) 𝑑 𝜇 (𝑥)

which is unique $𝜇$ -almost everywhere. Such an $𝑓$ is called the Radon-Nikodym derivative, denotes

𝑓 = 𝑑 𝜈 𝑑 𝜇

Proof

proof

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Radon-Nikodym theorem

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