Distribution function (measure theory)

Definitions

The first definition presented here is typically used in Analysis (harmonic analysis, Fourier Analysis, and integration theory in general) to analysis properties of functions.

Definition 1: Suppose ${\displaystyle (X,\mathcal{B},\mu )}$ is a measure space, and let ${\displaystyle f}$ be a real-valued measurable function. The distribution function associated with ${\displaystyle f}$ is the function ${\displaystyle d_f:[0,\mathrm{\infty })\to \mathbb{R}\cup \{\mathrm{\infty }\}}$ given by$${\displaystyle d_f(s)=\mu (\{x\in X:|f(x)|>s\})}$$It is convenient also to define ${\displaystyle d_f(\mathrm{\infty })=0}$.

The function ${\displaystyle d_f}$ provides information about the size of a measurable function ${\displaystyle f}$.

The next definitions of distribution function are straight generalizations of the notion of distribution functions (in the sense of probability theory).

Definition 2. Let ${\displaystyle \mu }$ be a finite measure on the space ${\displaystyle (\mathbb{R},\mathcal{B}(\mathbb{R}),\mu )}$ of real numbers, equipped with the Borel ${\displaystyle \sigma }$-algebra. The distribution function associated to ${\displaystyle \mu }$ is the function ${\displaystyle F_{\mu }:\mathbb{R}\to \mathbb{R}}$ defined by $${\displaystyle F_{\mu }(t)=\mu ((-\mathrm{\infty },t])}$$

It is well known result in measure theory that if ${\displaystyle F:\mathbb{R}\to \mathbb{R}}$ is a nondecreasing right continuous function, then the function ${\displaystyle \mu }$ defined on the collection of finite intervals of the form ${\displaystyle (a,b]}$ by $${\displaystyle \mu ((a,b])=F(b)-F(a)}$$ extends uniquely to a measure ${\displaystyle {\mu }_F}$ on a ${\displaystyle \sigma }$-algebra ${\displaystyle \mathcal{M}}$ that included the Borel sets. Furthermore, if two such functions ${\displaystyle F}$ and ${\displaystyle G}$ induce the same measure, i.e. ${\displaystyle {\mu }_F={\mu }_G}$, then ${\displaystyle F-G}$ is constant. Conversely, if ${\displaystyle \mu }$ is a measure on Borel subsets of the real line that is finite on compact sets, then the function ${\displaystyle F_{\mu }:\mathbb{R}\to \mathbb{R}}$ defined by $${\displaystyle F_{\mu }(t)=\{\begin{array}{ll}\mu ((0,t])& \text{if }t\ge 0\\ -\mu ((t,0])& \text{if }t<0\end{array}}$$ is a nondecreasing right-continuous function with ${\displaystyle F(0)=0}$ such that ${\displaystyle {\mu }_{F_{\mu }}=\mu }$.

This particular distribution function is well defined whether ${\displaystyle \mu }$ is finite or infinite; for this reason, a few authors also refer to ${\displaystyle F_{\mu }}$ as a distribution function of the measure ${\displaystyle \mu }$. That is:

Definition 3: Given the measure space ${\displaystyle (\mathbb{R},\mathcal{B}(\mathbb{R}),\mu )}$, if ${\displaystyle \mu }$ is finite on compact sets, then the nondecreasing right-continuous function ${\displaystyle F_{\mu }}$ with ${\displaystyle F_{\mu }(0)=0}$ such that $${\displaystyle \mu ((a,b])=F_{\mu }(b)-F_{\mu }(a)}$$ is called the canonical distribution function associated to ${\displaystyle \mu }$.