This question regards a process similar to that of Riemann integration.
Let $f$ be a continuous function on $[a,b]\subset\mathbb{R}$, $n$ a natural number and $\mathscr{P}_n:=\{x_0=a<x_1<...<x_{n-1}<x_n=b\}$ a partition of $[a,b]$. Let $t_{ji}$ be the $j$th set of points chosen randomly from the interval $[x_{i-1},x_i]$. (As in, the first trial is $j=1$, the second $j=2$, and so forth). Define the sum of the $j$th set of points to be $$S(j)=\sum_{i=1}^nf(t_{ji})(x_{i}-x_{i-1})$$
e.g. $S(1)=f(t_{11})(x_1-x_0)+f(t_{12})(x_2-x_1)+...+f(t_{1n})(x_n-x_{n-1})$,
and consider the average as we continue to sample indefinitely $$\lim_{k\rightarrow \infty} \frac{\sum_{j=1}^kS(j)}{k}$$
Is the following true: $$\lim_{k\rightarrow \infty}\frac{\sum_{j=1}^kS(j)}{k}=\int_{[a,b]}f(x)dx$$
Background: This was just something I thought about one day when learning Riemann integration and it slipped my mind until now. I don't have the ability to check this numerically for simple functions $f$ but I don't see why it wouldn't work.