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The coarea formula in its simple form states that $$ \int\limits_{A} f(x) \; dx = \int\limits_{\mathbb{R}} \int\limits_{g(x) = r} f(x) \; \omega \; dr $$ for a Gelfand-Leray form $\omega$ (i.e. $dg \wedge \omega = dx$), for any $f(x)$ and $g \colon A \to \mathbb{R}$, where $A \subset \mathbb{R}^n$, with appropriate properties. Is there a generalisation of this result to the case of an arbitrary measure on the left-hand side? In other words is there some similar representation for the integral $$ \int\limits_{A} f(x) \; \mu(dx) $$ where $\mu$ is not an absolutely continuous measure?

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Yes, as long as you deal with Polish spaces and Borel measurable mappings. It all boils down to Radon-Nicodim and the existence of a countable semiring generating the $\sigma$-algebra, so you can play the monotone class lemma to show that the representation you get for the measures of sets in the ring expands to the representation of integrals. The countability of the ring is needed to show that all those null sets in various a.e. still accumulate to a null set. – fedja Mar 10 '12 at 13:17

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