===== Drastic measures =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $F$ ... set |
| @#FFBB00: definiendum | @#FFBB00: $S$ in it |
| @#55EE55: postulate   | @#55EE55: $S:F\to{\mathbb K}\setminus\{0\}$ |
| @#55EE55: postulate   | @#55EE55: $S$ ... ${\mathbb K}$-linear |

>todo: ${\mathbb K}$-linear

-----

=== Discussion ===

"Normalization w.r.t. $S$",

$N_Sf:=(Sf)^{-1}\cdot f$,

has $SN_Sf=e$ and $[N_S]=[S^{-1}]$.

As $S$ is linear, 

$N_S(c\,f)=N_S(f)$

We'll also write

$\bar{f}:=(Sf)^{-1}\cdot f$ 

== Example 1 ==

For $F$ being a set of functions from ${\mathbb N}$ to some monoid for which a sum is defined that's always invertible, the general case (I think) is

$Sf:=\sum_{n=0}^\infty (L_nf)(n)$, 

where $(L_n)$ is a suitable sequence of linear operations (e.g. differential operators). 

For $L_n={\mathrm{id}}$ we get the standard sum (see below).

Explicitly, let $a$ be a sequence to $\mathbb C$ and the sum is non-zero, then
 
$\bar{a}:{\mathbb N}\to[0,1]$

$\bar{a}(n):=\frac{a(n)}{\sum_{k=0}^\infty a(k)}$

has

$\sum_{n=0}^\infty \bar{a}(n)=1$

== Example 1 ==

For ${\mathbb D}\subseteq{\mathbb R}^m$ we have integrals.

=== Reference ===
Wikipedia: [[https://en.wikipedia.org/wiki/Cumulative_distribution_function|Cumulative distribution function]]

-----
=== Subset of ===
[[ℝ valued function]]
=== Related ===
[[Probability space]]