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# Lebesgue decomposition theorem

Let $\mu$ and $\nu$ be two $\sigma$-finite signed measures in the measurable space $(\Omega,\mathscr{S})$. There exist two $\sigma$-finite signed measures $\nu_{0}$ and $\nu_{1}$ such that:

1. $\nu=\nu_{0}+\nu_{1}$;

2. $\nu_{0}\ll\mu$ (i.e. $\nu_{0}$ is absolutely continuous with respect to $\mu$;)

3. $\nu_{1}\perp\mu$ (i.e. $\nu_{1}$ and $\mu$ are singular.)

These two measures are uniquely determined.

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

28A12*no label found*

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## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

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new question: A good question by Ron Castillo

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new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

Jun 13

new question: young tableau and young projectors by zmth

Jun 11

new question: binomial coefficients: is this a known relation? by pfb