I guess the question isn't entirely precise. Given a sequence $\{x_i\}$ of sample points in $G$, drawn from a distribution $\mu$, I'm looking for a function which measures how "close" the empirical measure $\frac{1}{n} \sum_{i=1}^n \delta_{x_i}$ on $G$ is to the true measure $\mu$, and hoping that there is such a function which, when normalized, converges as $n\to \infty$ to a distribution which does not depend on $\mu$. Does that help?

Excellent example! Can anything be said if $F$ is smooth, bounded, and $\in L^{1+\epsilon}$ for all $\epsilon>0$? I am thinking of $F$ which are smooth, bounded, and decay like $F(t) \sim |t|^{-1}$ as $|t|\to \infty$.

The lower shriek is this case is just extension by zero, so $\Gamma(W, i_! \mathcal{O}_U) = \Gamma(W,\mathcal{O}_U)$ if $W\subset U$, and $0$ otherwise. So in this case $\Gamma(i_! \mathcal{O}_U) = \Gamma(i_! \mathcal{O}_V) = 0$, since $U$ and $V$ are strict subsets of $X$.