Step functions/Integral from above and from below/Section

A step function. In the statistic context, it is also called a histogram or a bar chart.

Definition

Let ${}I$ be a real interval with endpoints ${}a,b\in \mathbb {R}$. Then a function

$t\colon I\longrightarrow \mathbb {R}$

is called a step function, if there exists a partition

${}a=a_{0}<a_{1}<a_{2}<\cdots <a_{n-1}<a_{n}=b\,$

of ${}I$ such that ${}t$ is constant

on every open interval ${}]a_{i-1},a_{i}[$.

This definition does not require a certain value at the partition points. We call the interval ${}]a_{i-1},a_{i}[$ the ${}i$-th interval of the partition, and ${}a_{i}-a_{i-1}$ is called the length of this interval. If the lengths of all intervals are constant, then the partition is called an equidistant partition.

Definition

Let ${}I$ be a real interval with endpoints ${}a,b\in \mathbb {R}$, and let

$t\colon I\longrightarrow \mathbb {R}$

denote a step function for the partition ${}a=a_{0}<a_{1}<a_{2}<\cdots <a_{n-1}<a_{n}=b$, with the values ${}t_{i}$, ${}i=1,\ldots ,n$. Then

${}T:=\sum _{i=1}^{n}t_{i}(a_{i}-a_{i-1})\,$

is called the step integral of ${}t$ on ${}I$.

We denote the step integral also by ${}\int _{a}^{b}t(x)\,dx$. If we have an equidistant partition of interval length ${}{\frac {b-a}{n}}$, then the step integral equals ${}{\frac {b-a}{n}}{\left(\sum _{i=1}^{n}t_{i}\right)}$. The step integral does not depend on the partition chosen. As long as we have a step function with respect to the partition, one can pass to a refinement of the partition.

Definition

Let ${}I$ denote a bounded interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. Then a step function

$t\colon I\longrightarrow \mathbb {R}$

is called a step function from above for ${}f$, if ${}t(x)\geq f(x)$ holds for all ${}x\in I$. A step function

$s\colon I\longrightarrow \mathbb {R}$

is called a step function from below for ${}f$, if ${}s(x)\leq f(x)$ holds for all

${}x\in I$.

A step function from above (below) for ${}f$ exists if and only if ${}f$ is bounded from above (from below).

Definition

Let ${}I$ denote a bounded interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. For a step function from above

$t\colon I\longrightarrow \mathbb {R}$

of ${}f$, with respect to the partition ${}a_{i}$, ${}i=0,\ldots ,n$, and values ${}t_{i}$, ${}i=1,\ldots ,n$, the step integral

${}T:=\sum _{i=1}^{n}t_{i}{\left(a_{i}-a_{i-1}\right)}\,$

is called a step integral from above for ${}f$ on ${}I$.

Definition

Let ${}I$ denote a bounded interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. For a step function from below

$t\colon I\longrightarrow \mathbb {R}$

of ${}f$, with respect to the partition ${}a_{i}$, ${}i=0,\ldots ,n$, and values ${}s_{i}$, ${}i=1,\ldots ,n$, the step integral

${}S:=\sum _{i=1}^{n}s_{i}{\left(a_{i}-a_{i-1}\right)}\,$

is called a step integral from below for ${}f$ on ${}I$.

Different step functions from above yield different step integrals from above.

Definition

Let ${}I$ denote a bounded interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function, which is bounded from above. Then the infimum of all step integrals of step functions from above

of ${}f$ is called the upper integral of ${}f$.

Definition

Let ${}I$ denote a bounded interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function, which is bounded from below. Then the supremum of all step integrals of step functions from below

of ${}f$ is called the lower integral of ${}f$.

The boundedness from below makes sure that there exists at all a step function from below, so that the set of step integrals from below is not empty. This condition alone does not guarantee that a supremum exists. However, if the function is bounded from both sides, then the upper integral and the lower integral exist. If a partition is given, then there exists a smallest step function from above (a largest from below) which is given by the suprema (infima) of the function on the intervals of the partition. For a continuous function on a closed interval, these are maxima and minima. To compute the integral, we have to look at all step functions for all partitions.