Riemann sum without the boundary sets

Let $X\in \mathbb R^m$ be J-measurable set and $f:X\to \mathbb R$ an integrable function. I would like to prove this Riemann sum formula still holds

$$\int _Xf(x)dx=\lim_{|D|\to 0}\Sigma(f;D^*)$$

even we take only the sets of the decomposition $D^*$ which don't have common points with the boundary of $X$.

Remarks:

We define the Riemann sum as $\Sigma (f;D^*)=\sum_{i=1}^kf(\xi_i)volX_i$, where the decomposition $D^*=\{X_1,\ldots,X_k\}$ and $X_i$ are J-measurable with $\xi_i\in X_i$.

I want to prove if we restrict to only the $X_i$ without any point in commum with the boundary of $X$, then we can use only these sets in the Riemann sum getting the same result as if we would sum every $X_i$.

I can see this intuitively, but I don't know how we can prove it formally.