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The classical coarea formula provides us a possibility to reduce the integration over some set to the integration over the slices of this set. For example, we can reduce the integration over a unit ball to the integration over spheres: $$ \int\limits_{B(0,1)} f(x) \, dx = \int\limits_{0}^{1} \int\limits_{S(0,t)} f(x)\,dS \, dt. $$ I'm looking for the generalization of this result to the case of topological groups. For example, is there some similar formula that allows us to reduce the integral over $GL(n)$ to an integral over $O(n)$ and etc?

I'm also looking for such decomposition of integral over the additive group of symmetric matrices. I want to reduce the integration to the integration over the set of eigenvalues.

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Weil's formula for measures on homogeneous spaces would give such a reduction from $\operatorname{GL}{(n)}$ to $\operatorname{O}{(n)}$. See e.g. theorem 6. b) here – t.b. Apr 26 '12 at 13:24
See chapter 12, esp. section 4, of Lang's Real and Functional Analysis. – KCd Apr 26 '12 at 14:22
Perhaps you can find something on Federer's "Geometric Measure Theory". Is a really hardto-read book, but you can find lots of things in non-standard topic on measure theory there. – matgaio Apr 26 '12 at 17:57

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