Monte Carlo Integration

For the integral ${\int }_a^{b}g(x)dx<\infty$ For IID RV $X_i\in (a,b)$ with PDF $f(x)>0,\forall x\in (a,b)$ The integral can be approximated with $M_n=\frac{1}{n}{\sum }_{i=1}^n\frac{g(X_i)}{f(X_i)}$ Then, $E(M_n)={\int }_b^{a}g(x)dx$ by WLLN

Proof