Almost everywhere

The notion that a property is valid almost everywhere comes from measure theory , a branch of mathematics , and is a weakening of the fact that the property is valid for all elements of a set.

definition

Given is a dimension space and a property that can be meaningfully defined for all elements of . It is now said that the property holds almost everywhere (or -almost everywhere or for- almost all elements) if there is a -null set , so that all elements in the complement of the null set have the property. ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle E}$ ${\displaystyle \Omega }$ ${\displaystyle E}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle N}$ ${\displaystyle N^{C}}$

comment

It is important that the property can really be defined for all , i.e. the elements of the basic set. In addition, it is particularly not required that the amount on which does not apply is measurable. This quantity only has to be contained in a zero quantity. When the measurements are complete , both coincide. ${\displaystyle E}$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle E}$

Examples

Lebesgue measure

Let us consider the measure space as an example , that is, the closed unit interval from 0 to 1, provided with Borel's σ-algebra and the Lebesgue measure . If we now consider the sequence of functions ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda )}$

${\displaystyle f_{n}(x)=x^{n}}$,

so this converges to 0, on the point set it is constant 1. But since every point set is a Lebesgue null set, and the sequence of functions on the complement (in the measure space) converges 1 to 0, it converges almost everywhere to 0. ${\displaystyle [0,1)}$ ${\displaystyle \{1\}}$ ${\displaystyle \lambda }$

The Dirichlet function

${\displaystyle D(x)={\begin{cases}1,&{\mbox{wenn }}x{\mbox{ rational,}}\\0,&{\mbox{wenn }}x{\mbox{ irrational.}}\end{cases}}}$

-Almost everywhere on the unit interval is 0, because . ${\displaystyle \lambda }$ ${\displaystyle \lambda (\{x\in [0,1];D(x)\not =0\})=\lambda ([0,1]\cap \mathbb {Q} )=0}$

Dirac measure

We choose the same dimension space as above, this time with the Dirac dimension on the 1 ( ) . When examining the same sequence of functions, this measure gives exactly the opposite result: the interval is a zero set and the sequence of functions is constant on the set with measure 1. The sequence of functions is thus constant almost everywhere. ${\displaystyle \mu _{2}=\delta _{1}}$ ${\displaystyle [0,1)}$ ${\displaystyle \delta _{1}}$ ${\displaystyle \{1\}}$ ${\displaystyle \delta _{1}}$

The Dirichlet function is 1 almost everywhere, because . ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{1}(\{x\in [0,1];D(x)\not =1\})=\delta _{1}([0,1]\setminus \mathbb {Q} )=0}$

The choice and specification of the measure used is therefore essential for using the phrase “almost everywhere”.

Countable measure

For an arbitrary set there is a measure space, whereby for all it is defined: ${\displaystyle X}$ ${\displaystyle (X,{\mathcal {P}}(X),\mu _{\leq \aleph _{0}})}$ ${\displaystyle A\subseteq X}$

${\displaystyle \mu _{\leq \aleph _{0}}(A)={\begin{cases}0,&{\mbox{wenn }}A{\mbox{ abzählbar,}}\\\infty ,&{\mbox{wenn }}A{\mbox{ überabzählbar.}}\end{cases}}}$

The term “ -almost all” then means: For all elements, with the exception of at most a countable number. ${\displaystyle \mu _{\aleph _{0}}}$

A measure analogous to “almost all” with the meaning “for all elements with a finite number of exceptions” is not possible for dimensions. Such a function

${\displaystyle \mu _{<\aleph _{0}}(A)={\begin{cases}0,&{\mbox{wenn }}A{\mbox{ endlich,}}\\\infty ,&{\mbox{wenn }}A{\mbox{ unendlich,}}\end{cases}}}$

is not σ-additive for infinite . ${\displaystyle X}$

Pretty sure

In stochastics , a property on the probability space that applies almost everywhere is also referred to as an almost certain (or almost certain ) property. ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle P}$

application

As a typical and important application of the term presented here, we again consider the dimension space and a measurable function . ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda )}$ ${\displaystyle f\colon [0,1]\rightarrow \mathbb {R} }$

From     follows     almost everywhere. ${\displaystyle \int _{[0,1]}|f|\mathrm {d} \lambda =0}$ ${\displaystyle f=0}$

Proof: If it weren't for almost everywhere, there would and there would be a with . There follows , contrary to the premise. So it has to be almost everywhere. ${\displaystyle f=0}$ ${\displaystyle \textstyle 0<\lambda (\{x\in [0,1];f(x)\not =0\})=\lambda (\bigcup _{n\in \mathbb {N} }\{x\in [0,1];|f(x)|>{\tfrac {1}{n}}\})}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \textstyle 0<\lambda (\{x\in [0,1];|f(x)|>{\frac {1}{n}}\})}$ ${\displaystyle \textstyle |f|\geq {\tfrac {1}{n}}\chi _{\{x\in [0,1];|f(x)|>{\frac {1}{n}}\}}}$ ${\displaystyle \textstyle \int _{[0,1]}|f|\mathrm {d} \lambda \geq \int _{[0,1]}{\frac {1}{n}}\chi _{\{x\in [0,1];|f(x)|>{\frac {1}{n}}\}}\mathrm {d} \lambda ={\frac {1}{n}}\cdot \lambda (\{x\in [0,1];|f(x)|>{\tfrac {1}{n}}\})>0}$ ${\displaystyle f=0}$

See also

• Pointwise convergence μ-almost everywhere
• Uniform convergence μ-almost everywhere
• Almost all (with countably infinite basic sets)

literature

• Jürgen Elstrodt : Measure and integration theory. 4th, corrected edition. Springer, Berlin et al. 2005, ISBN 3-540-21390-2 .