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 $\Bbb R^{n}$. For $A\subset \Bbb S^{n-1}$ $$\mu(A)= \int_{\Bbb S^{n-1}}\mathcal{H}^{n-2}(A\cap\Bbb S_v^{n-2})d\sigma_{n-1}(v) $$ where $d\sigma_{n-1}$ represent the Lebesgue surface measure on $\Bbb S^{n-1}$ and $\mathcal{H}^{n-2}$ is the the $n-2$-Hausdorff measure.
Question: How can one prove that
$$\mu= |\Bbb S^{n-2}|\mathcal{H}^{n-1}¬\Bbb S_v^{n-1}:= |\Bbb S^{n-2}|\sigma_{n-1}$$
For further information also see this:How to prove that : $\mathcal{H}^{n-2}(\Bbb S_v^{n-2})=|\Bbb S^{n-2}|$