 Solution

Assume independent samples $X_i\sim g$, $\int f|\theta |<\infty$, and $g>0$ wherever $f\theta$ can contribute. The importance weight is $w(x)=f(x)/g(x)$, and the ordinary importance sampling estimator isIts expectation is $\int \theta (x)f(x)dx=\mu$, by cancellation of $g$ in the expectation integral. The strong law of large numbers gives almost-sure convergence to $\mu$. Common probability support is sufficient; the precise support condition for importance sampling only requires covering the nonzero integrand for this particular expectation.