Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} u(x+rz) dS(z). $$ (p. 26, Evans' PDE, 1st edition) where $u \in C^2 (U)$.

There are quite rough discussions about it in the same spirit, one by DG, and the other by Hausdorff measure. But both are quite rough to me and I'm having trouble following the steps.

So my questions are (1) both are really rigorous? (2) Will studying the textbooks they mention suffice? (for example, for DG approach, I may study Lee's SM and for Hausdorff measure approach I study Evans' measure textbook)

Also refer

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    $\begingroup$ To make the question more self-contained: what properties does $u$ have? $\endgroup$ – Chappers Jun 9 '15 at 17:38

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