Assume you have two functions $f,g: [a,b]\times [c,d]\rightarrow \mathbb{R}$.
Let $\Pi_1^n=(a=x_0,x_1,x_2,\ldots,x_n=b)$ be a partition of $[a,b]$ and let $\Pi_2^n$ be a partition of $[c,d]$.
Define $\Delta_2\Delta_1g(x_i,y_j)=g(x_{i},y_j)-g(x_i,y_{j-1})-g(x_{i-1},y_{j})+g(x_{i-1},y_{j-1})$.
Let $x_{i-1}\le\zeta_i\le x_i$,
$y_{j-1}\le\xi_j\le y_j $.
Do you know under what conditions the sums $\sum\limits_{i=1}^{|\Pi_1^n|-1}\sum\limits_{j=1}^{|\Pi_2^n|-1}f(\zeta_i,\xi _j)\Delta_2\Delta_1g(x_i,y_j)$
converges as both the norms of the partitions goes to zero as n goes to infinity?
Also assume now that $\frac{\partial^2}{\partial x\partial y}g(x,y)$ exist. Under what conditions does then the sum
$\sum\limits_{i=1}^{|\Pi_1^n|-1}\sum\limits_{j=1}^{|\Pi_2^n|-1}f(\zeta_i,\xi _j)\Delta_2\Delta_1g(x_i,y_j)$
converge to
$\int_a^b\int_c^df(x,y)\frac{\partial^2}{\partial x\partial y}g(x,y)dydx$?
If you have books talking about this, I would love to hear about them.