Let our sample space be the unit square, $E = [0,1]^2$. Consider the measurable space $(E,\mathcal{B})$ where $\mathcal{B}$ is the Borel sigma algebra on $E$.
Let $\mu_n$ be the finitely supported probability measure, defined by $\mu_n(\{x\})=\frac{1}{n^2}$ for all $x \in E$ of the form $x=(i/n,j/n)$, $i,j \in \{1,\cdots,n\}$, and $0$ otherwise.
I have to show that the sequence of measures $\mu_n$ converges setwise to the Lebesgue measure on $[0,1]^2$. Let us call it $\mu$.
My attempt: I have been able to show that $\mu_n$ converges to $\mu$ weakly, by showing that the CDF's converge pointwise, using the definition of the Reimann integral as the limit of Reimann sums. But I'm getting confused showing that the area of any set converges.