Usually the Riemann integral for $\mathbb{R^n}$ is defined on a hyperrectangular region $T$, by partitioning the region's "edges", which are $(a_1,b_1) \times (a_2,b_2) \times \dots \times (a_n,b_n)$ where $T$ is partitioned into $m$ smaller,disjoint sub-hyperrectangles such that $T=C_1 \cup \dots \cup C_m$. Let's assume that we have a continuous function $f:T \to \mathbb{R}$. When we name the largest edge of these small $C_k$ rectangles as $\delta$ then Riemann integral is defined as:
$$ \int_{T}f(x_1,\dots,x_n)dV= \lim_{\delta \to 0}\sum_{k=0}^{m}f(P_k)m(C_k) $$
where $P_k$ is a point in $C_k$ and $m(C_k)$ is the volume of $C_k$, if this limit exists.
How can this definition be extended to a nonrectangular, closed and bounded subset $\Omega$ of $\mathbb{R^n}$? I fail to find a satisfactory definition anywhere. I think of such a procedure (in a heuristic way): Find the tightest bounding, axis-aligned hyperrectangle $T_{\Omega}$ for $\Omega$. Then execute the regular partitioning mechanism for $T_{\Omega}$ again; but this time I define a function $I(P)$ which returns $1$ when $P \in \Omega$ and $0$ otherwise. Then the following should give the integral of a continuous function $f$ which is defined on $\Omega$; as:
$$ \int_{\Omega}f(x_1,\dots,x_n)dV = \lim_{\delta \to 0}\sum_{k=0}^{m}I(P_k)f(P_k)m(C_k)$$
Is this definition true? How can one make it correct and rigorous if not?