Uniform convergence μ-almost everywhere

The uniform convergence μ-almost everywhere is a convergence term in measure theory for function sequences . It will also ℒ ^∞ convergence, or convergence in ℒ ^∞ called because of convergence with respect to the corresponding norm. Thus, the uniform convergence μ-almost everywhere is both a borderline case of the convergence in the p-th mean and a weakening of the uniform convergence . There are other convergence terms with the addition “almost everywhere”, such as point-wise convergence μ-almost everywhere . To avoid confusion, the full name of the concept of convergence should always be used. If only the “convergence almost everywhere” is spoken of, the point-wise convergence is usually meant almost everywhere. Likewise, the uniform convergence μ-almost everywhere should not be confused with the almost uniform convergence , this is a weaker concept of convergence. ${\displaystyle {\mathcal {L}}^{\infty }}$

definition

Two different definitions can be given, one using the norm and one using uniform convergence. Both definitions are equivalent. Let a dimension space and measurable functions be given ${\displaystyle {\mathcal {L}}^{\infty }}$ ${\displaystyle (X,{\mathcal {A}},\mu )}$

${\displaystyle f\colon X\to \mathbb {K} ,(f_{n}\colon X\to \mathbb {K} )_{n\in \mathbb {N} }}$.

Uniform convergence μ-almost everywhere

The sequence of functions is called μ-almost everywhere uniformly convergent , if a μ-null set exists, so that on the complement of , i.e. on , converges uniformly to . So it applies ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle f_{n}}$ ${\displaystyle A}$ ${\displaystyle X\setminus A}$ ${\displaystyle f}$

${\displaystyle \lim _{n\rightarrow \infty }\,\sup _{x\in X\setminus A}\left|f_{n}(x)-f(x)\right|=0.}$

Convergence in ℒ ^∞

The semi- norm defined by the essential supremum is given

${\displaystyle \Vert f\Vert _{{\mathcal {L}}^{\infty }}:=\mathrm {ess} \sup |f|}$.

Then the sequence of functions is called convergent in if ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {L}}^{\infty }}$

${\displaystyle \lim _{n\to \infty }\Vert f_{n}-f\Vert _{{\mathcal {L}}^{\infty }}=0}$

is.

Relationship to other convergence terms

Almost uniform convergence

The almost uniform convergence follows from the uniform convergence μ-almost everywhere . This calls for uniform convergence on a set of arbitrarily small dimensions. Since with the uniform convergence μ-almost everywhere there is always uniform convergence with the exception of a zero set, this is always fulfilled.

The reverse does not apply: For example, on the measurement space the sequence of functions for any small amount on the interval converges uniformly to 0, so that it also converges almost uniformly to 0 on the interval . However, the sequence of functions does not converge evenly μ-almost everywhere. ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda |_{[0,1]})}$ ${\displaystyle f_{n}(x)=x^{n}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle [0,1-\epsilon ]}$ ${\displaystyle [0,1]}$

Convergence in the pth mean

In case of a finite measure space follows from the uniform convergence of μ-almost everywhere convergence in pth mean to , because by the Hölder's inequality , one can show that ${\displaystyle p\in (0,\infty )}$

${\displaystyle \|f\|_{p}\leq \mu (X)^{1/p}\|f\|_{_{\infty }}}$.

applies. However, this conclusion is generally wrong for non-finite measure spaces. For example, if you define the sequence of functions

${\displaystyle f_{n}(x)={\tfrac {1}{n}}\chi _{[0,n]}(x)}$

on so is ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),\lambda )}$

${\displaystyle \lim _{n\to \infty }\|f_{n}\|_{\infty }=\lim _{n\to \infty }{\tfrac {1}{n}}=0{\text{ aber }}\lim _{n\to \infty }\|f_{n}\|_{1}=1}$.

The conclusion from convergence in the p-th mean to uniform convergence almost everywhere is generally wrong both in finite measure spaces and in general measure spaces. The sequence of functions on the finite measure space converges, for example, for the p-th mean to 0, but not almost everywhere evenly to 0. ${\displaystyle f_{n}(x)=x^{n}}$ ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda )}$ ${\displaystyle p\in [1,\infty )}$

Point by point almost everywhere, made to measure and locally made to measure

Since the point-wise convergence μ-almost everywhere as well as the convergence to measure and the convergence locally to measure follow from the almost uniform convergence, all three types of convergence already follow after the above section from the uniform convergence μ-almost everywhere.

General formulation

The uniform convergence almost everywhere can be defined analogously for functions with values ​​in a metric space . a sequence of functions is then called uniformly convergent almost everywhere if there is a null set such that ${\displaystyle (M,d)}$ ${\displaystyle A}$

${\displaystyle \lim _{n\to \infty }\sup _{x\in X\setminus A}\ d(f_{n}(x),f(x))=0}$

applies.

literature

• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .