Definition:Integral Sign

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Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let:

$\displaystyle \int f \rd \mu := \sup \set {\map {I_\mu} g: g \le f, g \in \EE^+}$

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

$\MM_{\overline \R}^+$ 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
$\map {I_\mu} g$ denotes the $\mu$-integral of the positive simple function $g$
$g \le f$ denotes pointwise inequality
$\EE^+$ 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.