Let $p_1$ and $p_0$ be two probability density functions and let the function defined by $$l(y)=\frac{p_1(y)}{p_0(y)}$$ be the probability ratio function.
$$\int_{\{y:\,l(y)\leq t\}} l(y)p_0(y)\,\mathrm{d}y=E_{P_0}[l(Y)\mid l\leq t]$$
I wrote this when I asked a question and I think this equality is wrong. I have just realized it because there is no normalization.
In other words, I have another equation:
$$\frac{1}{P_0[a\leq l(Y)\leq b]}\int_{a\leq l(y)\leq b} l(y) p_0(y) \, \mathrm{d}y = E_{P_0}[l(Y)\mid a\leq l\leq b] $$
which seems to me correct, beacause there is normalization. Am I right?