From Wikipedia
Let $X:\Omega\to \mathbb{R}$ be a random variable in some probability space $(\Omega,\mathcal{F},P)$. The basic idea of importance sampling is to change the probability $P$ so that the estimation of $E[X;P]$ is easier. Choose a random variable $L\geq 0$ such that $E[L;P]=1$ and that $P$-almost everywhere $L(\omega)\neq 0$. The variate $L$ defines another probability $P^{(L)}=L\, P$ that satisfies $ \mathbf{E}[X;P] = \mathbf{E}\left[\frac{X}{L};P^{(L)}\right]. $
I was wondering if $P^{(L)}$ is a probability measure on $\mathbb{R}$ induced by $L$ from $\Omega$?
How shall $\mathbf{E}\left[\frac{X}{L};P^{(L)}\right]$ be understood as an integral?
Why is it true that $ \mathbf{E}[X;P] = \mathbf{E}\left[\frac{X}{L};P^{(L)}\right]$?
Thanks!