a $d$-sheeted covering map, with $d<+\infty$.

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\[T:Weylbianhuan\] Suppose conditions (a), (b), (c), and (e) hold. Then we have $$\label{F:Weylbianhuan}
\int_X f(x)
dx=\frac{1}{d}\int_Y\left(\int_{G/K}f(\sigma_g(y))d\mu([g])\right)
J(y)dy$$ for all $f\in C^\infty(X)$ with $f\geq0$ or with $f\in L^1(X,dx)$, where $J\in C^\infty(Y)$ is determined by Formula (\[F:J\]).

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By Proposition \[P:localdiffeo\], $\varphi: G/K\times
Y'\rightarrow X'$ is a local diffeomorphism. By the covering condition (e), $\varphi$ is a $d$-sheeted covering map. So by Proposition \[P:jifenbianhuan\], for $f\in C^\infty(X)$ with $f\geq0$ or $f\in L^1(X,dx)$, we have $$\begin{aligned}
\int_X f(x)dx
=&\int_{X'} f(x)dx\\
=&\frac{1}{d}\int_{G/K\times Y'}f(\varphi([g],y))\varphi^*(dx)\\
=&\frac{1}{d}\int_{G/K\times Y'}f(\sigma_g(y))J(y)d\mu([g])dy\\
=&\frac{1}{d}\int_{Y'}\left(\int_{G/K}f(\sigma_g(y))d\mu([g])\right)J(y)dy\\
=&\frac{1}{d}\int_Y\left(\int_{G/K}f(\sigma_g(y))d\mu([g])\right)J(y)dy\end{aligned}$$

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\[C:leihanshubianhuan\] Under the same conditions as in the above Theorem, if furthermore $f(\sigma_g(x))=f(x), \forall g\in G, x\in X$, then $$\label{F:leihanshubianhuan}
\int_X f(x) dx=\frac{\mu(G/K)}{d}\int_Y f(y)J(y)dy.$$

0.3cm To make the above conclusion more available, we give some criterions on when the map $\varphi: G/K\times Y'\rightarrow
X'$ is a covering map.

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\[P:covering\] Let $M,N$ be smooth $n$-dimensional manifolds. Then an everywhere regular smooth map $\varphi : M\rightarrow N$ is a $d$-sheeted covering map if and only if for each $y\in N$, $\varphi^{-1}(y)$ has $d$ points.

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0.2cm The “$\Rightarrow$" part is obvious. We prove the “$\Leftarrow$" part.

0.3cm For $y\in N$, let $\varphi^{-1}(y)=\{x_1,\cdots,x_d\}$. Since $\varphi$ is everywhere regular, there exists open 