Assume $Z_1, Z_2, Z_3,...$ are independent and identically distributed R>V.s with with density g>0. Prove that for all continuous function f s.t. $\int_R|f|<\infty$, $\frac 1 i \sum_{n=1}^i \frac {f(Z_n)} {g(Z_n)} \to \int_R|f|$ almost surely as $i \to \infty$.
Since $\int_R|f|<\infty$, do i need to use borel cantelli lemma to show that probability of some event equal to 0 first? I am a little stuck on how to tackle the convergence of the ratio btween f and g part. Any hints wouold help thanks.
This is just the strong law of large numbers. $\frac 1 i \sum_{n=1}^{i} \frac {f(X_n)} {g(X_n)} \to E\frac {f(X_1)} {g(X_1)} $ almost surely and $E\frac {f(X_1)} {g(X_1)} =\int \frac {f(x)} {g(x)} g(x) dx=\int f(x) dx$. The limit is $\int f(x)dx$ not $\int |f(x)| dx$. [Also $Z_n$ should be $X_n$ in the statement].