I met an equality in the book "Harmonic analysis and approximation on the unit sphere" by Wang Kunyang, in page 23. I cannot follow the reasoning below,
\begin{align} \int_{\Omega_{n}} P^{n}_{k}(\xi\cdot\eta)P^{n}_{j}(\xi\cdot\eta)d\omega_{n}(\eta)=\int_{[-1,1]}\int_{\Omega_{n-1}}P^{n}_{k}(t)P^{n}_{j}(t)d\omega_{n-1}d\mu(t) \end{align}
where $\Omega_{n}$ is the unit sphere in $R^{n}$, $\xi$ is in $\Omega_{n}$ and fixed, $P^{n}_{k}$ is a polynomial on $[-1,1]$ and $\mu$ is a measure defined on $\mathscr{E}$ which is a collection of Lebesgue measurable sets included in $[-1,1]$. $\mu$ is defined as $$\mu(E)=\int_{[-1,1]}\chi_{E}(t)(1-t^{2})^{\frac{n-3}{2}}dt, \quad\quad E\in\mathscr{E}$$
where $\chi_{E}$ is the characteristic function of $E$.
I think the precise definition of $\mu$ does not matter here. My problem is I don't understand how we can derive the right-hand side from the left-hand side of the equality. Whay $\mu$ is defined this way and how does the $\mu$ come in the reasoning?:-(