Convergence of $L^p$ of approximation Let $f \in L^p(\mathbb R^n)$ be given. Consider a partition of rectangles $I_{ij}:=[x_i,x_{i+1}]\times [x_j,x_{j+1}]$ of $\mathbb R^2.$
Then, we may define the coefficients 
$$\alpha_{ij}= \frac{1}{\left\lvert I_{ij} \right\rvert} \int_{I_{ij}} f(s) \ ds$$
and consider the function $$g(x):=\sum_{ij} \alpha_{ij}\chi_{I_{ij}}(x).$$
I ask: If the distance between $x_i$ and $x_{i+1}$ goes uniformly in $i$ to zero. Does this imply that $g$ converges in $L^p$ to $f$? The functions $\chi$ are just indicator functions.
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The answer is yes, at least for $p\ge1$. This follows because compactly supported continuous functions are dense in $L^p$ and such functions are uniformly continuous. 
Here are details. For any $f\in L^p(\R^n)$, let $\bar f$ be defined as your function $g$: 
\begin{equation*}
 \bar f:=\sum_{ij} \al_{f;ij}\chi_{I_{ij}}, 
\end{equation*}
where 
\begin{equation*}
 \al_{f;ij}:=\frac1{|I_{ij}|} \int_{I_{ij}} f(s)\,ds. 
\end{equation*}
Take any real $\ep\in(0,1)$ and take any continuous function $f_\ep$ with a support $K_\ep\subseteq[-M_\ep,M_\ep]^n$ for some real $M_\ep>0$ such that 
\begin{equation*}
 \|f-f_\ep\|_p<\ep. \tag{1}
\end{equation*}
Let $$\ep_1:=\ep/(2M_\ep+2)^{n/p}.$$ 
By the uniform continuity of $f_\ep$, there is some real $\de_\ep>0$ such that 
\begin{equation*}
 |f_\ep-\bar{f_\ep}|=\sum_{ij} |f_\ep-\bar{f_\ep}|\,\chi_{I_{ij}}\le
 \sum_{ij} \ep_1\chi_{I_{ij}}=\ep_1
\end{equation*}
as soon as 
\begin{equation*}
 \max_i(x_{i+1}-x_i)<\de_\ep. \tag{2}
\end{equation*}
Hence, 
\begin{multline*}
 \|f_\ep-\bar{f_\ep}\|_p^p
 =\int_{\R^n} |f_\ep-\bar{f_\ep}|^p
  =\int_{[-M_\ep-1,M_\ep+1]^n} |f_\ep-\bar{f_\ep}|^p \\ 
 \le\ep_1^p(2M_\ep+2)^n=\ep^p. \tag{3}
\end{multline*}
Also, for $h:=f-f_\ep$, by Jensen's inequality and (1),
\begin{multline*}
 \|\bar f-\bar{f_\ep}\|_p^p=\|\,\bar h\,\|_p^p=\int_{\R^n}|\,\bar h\,|^p
 =\sum_{ij}\int_{I_{ij}}|\,\bar h\,|^p
  =\sum_{ij}|\,\bar h\,|^p\,|I_{ij}| \\ 
  \le\sum_{ij}\bar{|h|^p}\,|I_{ij}|
  =\sum_{ij}\int_{I_{ij}}|h|^p
  =\|h\|_p^p=\|f-f_\ep\|_p^p<\ep^p. \tag{4}
\end{multline*}
Thus, by (1), (3), (4), and Minkowski's inequality, 
\begin{equation*}
 \|f-\bar f\|_p\le3\ep
\end{equation*}
as soon as (2) holds. 
