Given a subdivision $\{x_{0},x_{1},\ldots,x_{m}\}$ of $[a,b]$, the "$\Delta$-approximant" of a function $f$ is defined as the piecewise constant function $\phi$ taking the value \begin{equation} \phi(x) = \frac{1}{x_{k+1}-x_{k}}\int_{x_{k}}^{x_{k+1}}f(t)dt \end{equation} for all $x\in(x_{k},x_{k+1})$.
On page 129 in Real Analysis (Third Edition) by HL Royden, the author shows convergence in $L^{p}[a,b]$-norm of the $\Delta$-approximant of a function $f\in L^{p}[a,b]$ to this function as the subdivision is refined.
As far as I can tell, looking at the proof in the cited book, it's straightforward to generalize this result to bounded sets in $\mathbb{R}^{n}$. Nevertheless it would be nice to have a reference for this case (I've seen the result described as "well-known", but I haven't found it in any literature).