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In the Riemann Integral, how can I interpret the step function when it is on a partition of not $\mathbb{R}$, but $\mathbb{R^n}$?

Normally for step functions defined for the Riemann Integral, it is on a closed interval of $\mathbb{R}$. However, I am working with a definition that generalizes to $\mathbb{R^n}$. Specifically, Let $R = [a_1,b_1] \times ... \times [a_n,b_n]$. Then, I define a partition $P$ of $R$ as the collection of closed intervals $I ⊂ R$ obtained by partitioning each of the edges of $R$; thus for each $j = 1,...,n$ we select points $a_j = t_{j,0} < t_{j,1} < ··· < t_{j,N_j} = b_j$ and then

$P = \{[t_{1,i_{1−1}},t_{1,i_1}] × [t_{2,i_2−1},t_{2,i_2}] \times \cdots \times [t_{n,i_n−1},t_{n,i_n}] : i_j \in \{1,...,N_j\}$ for each $j = 1,...,n\}$.

An interval $I$ is defined to be $I = [t_{1,i_{1−1}},t_{1,i_1}] × [t_{2,i_2−1},t_{2,i_2}] \times \cdots \times [t_{n,i_n−1},t_{n,i_n}]$ with $\hat{I} = (t_{1,i_{1−1}},t_{1,i_1}) × (t_{2,i_2−1},t_{2,i_2}) \times \cdots \times (t_{n,i_n−1},t_{n,i_n})$

My definition of a step function is:

$\phi$ is a step function on $R$ if it is bounded and there is a partition $P$ of $R$ such that for each $I \in P$ there is a real constant $a_I$ such that $\phi ≡ a_I$ on $\hat{I}$. Therefore we have:

$$ \phi = \sum_{I \in P}{a_I \chi_\hat{I}} \space \space \text{on} \space \space \cup_{I \in P}\hat{I} $$

where we use the notation that, for any set $A ⊂ \mathbb{R^n}$, $\chi_A$ denotes the indicator function of $A$; thus $\chi_A(x) = 1$ if $x ∈ A$ and $\chi_A(x) = 0$ if $x ∈ \mathbb{R^n} \setminus A$.

Basically, I am not sure how the summation works within the step function. Does anyone know of an enlightening example? Does it have value on only one term and zeros everywhere else? Also, why would the partition $P$ be defined as such above when we are working in $\mathbb{R^n}$? Thanks!

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You pick a block in $\Bbb R^n$, call it $\bf B$. Now, you partition it into subblocks, ${\bf B}_1,\ldots,{\bf B}_\ell$. The characteristic function of the $i$-th block, call it $\chi_i$, is such that $\chi_i=1$ on ${\bf B}_i$ and $0$ elsewhere. For example, in $\Bbb R^2$, your blocks are rectangles, and your characteristic functions are rectangles of fixed height $=1$. Now, the function $s=\sum_{i=1}^\ell c_i\chi_i$ will take the value $c_i$ in ${\bf B}_i$ and $0$ elsewhere, and $\int_{\bf B}\chi_i={\rm vol}({\bf B}_i)$, so $$\int_{\bf B} s=\sum_{i=1}^{\ell}c_i{\rm vol}({\bf B}_i)$$

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