Fix $n \in \mathbb{N}$. Let $I = [0, 1]$, and for every $x \in [\frac{k}{n}, \frac{k + 1}{n})$ define the measure $$\nu_x(y) = \frac{1}{n}\sum_{i=0}^{n-1} \delta(y - (x - k/n + i / n)).$$ To see what $\nu_x$ looks like, note that $y = x - k/n + i/n$ is the equation of the straight line joining the points $(k/n, i/n)$ and $((k+1)/n, (i+1)/n)$.
For any integrable $f : [0, 1] \times [0, 1] \to \mathbb{R}$, we define $$I_n(f) = \int_0^1 \int_0^1 f(x, y) d\nu_x(y) dy.$$
Now, does this converge to the Lebesgue measure of $f$ over the unit square?
I am very interested in learning how to attack situations like this.