Definition:Integral Sign

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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.