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We know that in $\mathbb{R}^n$, we have transformation rules such as:

$\int_{\mathbb{R}^n}f(x-h) dx=\int_{\mathbb{R}^n}f(x) dx$

$\delta^n \int_{\mathbb{R}^n}f(\delta x) dx=\int_{\mathbb{R}^n}f(x) dx$.

The proof of such formulas can be easily found in a book about real analysis. They reduce the problem into an integral of a characteristic function of a measurable set and then use some properties of measure to prove it.

My problem is: what are the similar rules for integral on n-dimensional sphere?

For example,

What is the relationship between $\int_{\partial B(0,r)}f(x) dS_r$ and $\int_{\partial B(0,1)}f(rx)dS_1$, where $S_r$ is the area element of $\partial B(0,r)$? If there are some, how to prove it? (In addition, how to define such an integral using abstract integration theory?)

And why is $\int_{\partial B(0,r)}f(x) dS_r=\int_{\partial B(y,r)}f(x-y)dS_r$? ( This may be easy to imagine when the dimension is 1 or 2, but I want to know a proof starting from the definition of such a integral especially when the dimension is high.)


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up vote 2 down vote accepted

The "area element" on a $d$-dimensional manifold can be considered as $d$-dimensional Hausdorff measure. From the definition of Hausdorff measure, any isometry will preserve Hausdorff measure, while a map that multiplies all distances by a factor of $r$ will multiply $d$-dimensional Hausdorff measures by $r^d$. That is, if $T$ is a transformation from $M_1$ onto $M_2$ such that ${\rm dist}(T(x), T(y)) = r\, {\rm dist}(x,y)$ for all $x, y \in M_1$, and $m_1$ and $m_2$ are $d$-dimensional Hausdorff measure on $M_1$ and $M_2$ respectively, then for any $m_1$-measurable $E \subseteq M_1$, $m_2(T(E)) = r^d m_1(E)$, and for any $m_2$-integrable function $f$ on $M_2$, $\int f(T(x)) \, dm_1(x) = r^{-d} \int f(x)\, dm_2(x)$.

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