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Let $(X,M,\mu)$ be a measure space. Let $f\geq 0$ be integrable over $X$ wrt to $\mu$. How can I find the Lebesgue decomposition wrt $\mu$ of the measure $\lambda(E) = \int_E fd\mu $ for every $E\in M$.

So I know I must find measures $\lambda_1,\lambda_2$ such that $\lambda = \lambda_1+\lambda_2$, with $\lambda_1 \perp \mu$ and $\lambda_2 \ll \mu.$ Please how do I proceed?

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    The decomposition you are looking for is $\lambda=0+\lambda$.2012-04-09

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