 Solution

Assume the necessary support condition for importance sampling, $p\ll q$; the source's density assumptions alone do not ensure it. Put $\mu =E_{p}f$ and $w=p/q=(Z_q/Z_p)\tilde{p}/\tilde{q}$. The importance sampling estimator isIntegration under $q$ gives $E_q[wf]=\int pf=\mu$, proving unbiasedness. Independence givesThis variance is finite precisely when the first integral is finite. The assumption $E_p{f}^2<\infty$ ensures target square integrability, but need not ensure square integrability after importance reweighting; the variance can be infinite.