Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 1 down vote accepted

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 $$\int_{\bf B} s=\sum_{i=1}^{\ell}c_i{\rm vol}({\bf B}_i)$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.