Usually Monte Carlo method is used to compute integration. For example, let $g(x,\theta)$ be a continuous function about $x$ and $\theta$, $f(x \mid \theta)$ is a continuous pdf with parameter $\theta$. if the integral $E[g(x,\theta)]=\int g(x,\theta)f(x\mid \theta)dx$ has no closed form . Then we can use Monte Carlo approximation $E[g(x,\theta)] \approx \frac{1}{m}\sum_{l=1}^m g(X_i,\theta)$, where $X_1,\dots,X_m$ is a random sample from distribution $f(x \mid \theta)$. By law of large number, $\frac{1}{m}\sum_{l=1}^m g(X_i,\theta)$ converges to $E[g(x,\theta)]$ in probability for each $\theta \in \Theta$. Do we have theorem talking about uniform convergence of Monte Carlo approximation? I mean, under what condition, we have
$\sup\limits_{\theta\in\Theta} \left\| \frac1n\sum_{i=1}^n g(X_i,\theta) - \operatorname{E}[g(X,\theta)] \right\| \xrightarrow \ 0$ a.s. or in probability
Assume parameter space $\Theta$ is compact