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 GelfandLeray 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 lefthand 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?
