# Definition:Integral Sign

## Definition

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let:

- $\displaystyle \int f \rd \mu := \sup \, \left\{{I_\mu \left({g}\right): g \le f, g \in \mathcal E^+}\right\}$

denote the $\mu$-integral of the positive measurable function $f$. where:

- $\mathcal M_{\overline \R}^+$ denotes the space of positive $\Sigma$-measurable functions
- $\overline \R_{\ge 0}$ denotes the positive extended real numbers
- $\sup$ is a supremum in the extended real ordering
- $I_\mu \left({g}\right)$ denotes the $\mu$-integral of the positive simple function $g$
- $g \le f$ denotes pointwise inequality
- $\mathcal E^+$ denotes the space of positive simple functions

The symbol:

- $\displaystyle \int \ldots \rd \mu$

is called the **integral sign**.

Note that there are two parts to this symbol, which embrace the function $f$ which is being integrated.

## Historical Note

The integral sign $\displaystyle \int \ldots \rd x$ originated with Gottfried Wilhelm von Leibniz

In a manuscript dated $29$th October $1675$ he introduced a long letter $S$ to suggest the first letter of the word **summa** (Latin for **sum**).

At the time he was using the notation $\operatorname {omn} l$ (that is: **omnes lineae**, meaning **all lines**).

Then he noted:

*It will be useful to write $\int$ for $\operatorname {omn}$, thus $\int \, l$ for $\operatorname {omn} l$, that is, the sum of those $l$'s.*

At the same time he introduced the differential symbol $\rd$.

Thus he was soon writing $\rd x$, $\rd y$, and $\int \ldots \rd x$ soon followed.

In his $1684$ article *Nova Methodus pro Maximis et Minimis*, published in *Acta Eruditorum*, he casually drops the notation in place with very little explanation.