Let $n\ge 3$ and $\Bbb S^{n-1}$ denote the $n-1$-dimensional unit sphere in $\Bbb R^{n}$ namely, $\Bbb S^{n-1}= \{x \in \Bbb R^{n}: \|x\|=1\}$
For $v\in \Bbb S^{n-1}$ fixed we define $$\Bbb S_v^{n-2}=\Bbb S^{n-1}\cap (\Bbb R v)^\bot$$ where $(\Bbb R v)^\bot$ is the orthogonal of $v$ in $\Bbb R^{n}$.
Question: Let $f:\Bbb S^{n-1}\to \Bbb R$ be a measurable function. How to prove that
$$\int_{\Bbb S^{n-1}}f(x)d\sigma_{n-1}(x)= \int_{\Bbb S^{n-1}}\left(\int_{\Bbb S_v^{n-2}}f(y)\frac{d\sigma_{n-2}(y)}{|\Bbb S^{n-2}|}\right)d\sigma_{n-1}(v) $$
where $d\sigma_{n-1}$ represent the Lebesgue surface measure on $\Bbb S^{n-1}.$
Can anyone help?
For further information also see this:How to prove that : $\mathcal{H}^{n-2}(\Bbb S_v^{n-2})=|\Bbb S^{n-2}|$