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.)
Thanks.