I meet one problem on the probability and statistic theory.
"Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions $f_i:X\rightarrow\mathbb{R}_+$, $i=1,2$, such that, $$\mu_i(E)=\int_Ef_i(x)\mathrm{d}\lambda(x),\quad \forall E\in S.$$
$H_i$, $i=1,2$, is the hypothesis that $x$ is from the statistical population with probabilty measure $\mu_i$.
Then, we can derive the following relations from Bayes' Theorem, $$P(H_i|x)=\frac{P(H_i)f_i(x)}{P(H_1)f_1(x)+P(H_2)f_2(x)},\quad i=1,2.$$ "
I know the classic Bayes' Theorem, but I am confused about the last equality. How to deduce the last equality by Bayes' Theorem ?