# Step functions/Integral from above and from below/Section

## Definition

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

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

is called a step function, if there exists a partition

${\displaystyle {}a=a_{0}

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

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

This definition does not require a certain value at the partition points. We call the interval ${\displaystyle {}]a_{i-1},a_{i}[}$ the ${\displaystyle {}i}$-th interval of the partition, and ${\displaystyle {}a_{i}-a_{i-1}}$ 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 ${\displaystyle {}I}$ be a real interval with endpoints ${\displaystyle {}a,b\in \mathbb {R} }$, and let

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

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

${\displaystyle {}T:=\sum _{i=1}^{n}t_{i}(a_{i}-a_{i-1})\,}$
is called the step integral of ${\displaystyle {}t}$ on ${\displaystyle {}I}$.

We denote the step integral also by ${\displaystyle {}\int _{a}^{b}t(x)\,dx}$. If we have an equidistant partition of interval length ${\displaystyle {}{\frac {b-a}{n}}}$, then the step integral equals ${\displaystyle {}{\frac {b-a}{n}}{\left(\sum _{i=1}^{n}t_{i}\right)}}$. 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 ${\displaystyle {}I}$ denote a bounded interval, and let

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

denote a function. Then a step function

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

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

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

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

${\displaystyle {}x\in I}$.

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

## Definition

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

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

denote a function. For a step function from above

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

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

${\displaystyle {}T:=\sum _{i=1}^{n}t_{i}{\left(a_{i}-a_{i-1}\right)}\,}$
is called a step integral from above for ${\displaystyle {}f}$ on ${\displaystyle {}I}$.

## Definition

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

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

denote a function. For a step function from below

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

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

${\displaystyle {}S:=\sum _{i=1}^{n}s_{i}{\left(a_{i}-a_{i-1}\right)}\,}$
is called a step integral from below for ${\displaystyle {}f}$ on ${\displaystyle {}I}$.

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

## Definition

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

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

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

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

## Definition

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

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

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

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

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.