Pointwise convergence μ-almost everywhere

The point-wise convergence μ-almost everywhere , sometimes also called convergence μ-almost everywhere for short , is a convergence term in measure theory for function sequences . It corresponds to the point-wise convergence on the entire basic set with the exception of a μ- zero set , which corresponds to the mass-theoretical manner of speaking μ- almost everywhere . The μ stands for the measure used . If this is referred to differently, the letter is adapted accordingly. For the Lebesgue measure , for example, one would then speak of the point-wise convergence λ-almost everywhere. If it is clear which dimension is involved, the specification is dispensed with, one simply speaks of point- by- point convergence almost everywhere or convergence almost everywhere . It should be noted that there are other combinations of convergence terms and the way of speaking "almost everywhere", such as the uniform convergence μ-almost everywhere . Seen in this way, the term “convergence almost everywhere” is ambiguous, but in most cases it describes point-by-point convergence almost everywhere.

The probabilistic counterpart of point-wise convergence μ-almost everywhere is P-almost certain convergence .

definition

Let a dimension space and measurable functions be given . Then the sequence of functions is called point-wise convergent μ-almost everywhere against , if there is a set such that is and the sequence of functions converges on the complement of the set , i.e. on point-wise against . ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle f,(f_{n})_{n\in \mathbb {N} }\colon X\to \mathbb {K} }$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle \mu (A)=0}$ ${\displaystyle A}$ ${\displaystyle X\setminus A}$ ${\displaystyle f}$

example

Consider the dimension space and the sequence of functions ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),\lambda )}$

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

It converges point by point λ-almost everywhere to 0, because the sine only takes values ​​between −1 and 1. All values ​​in the interval become smaller and smaller when exponentiated with larger ones and approach 0. These values ​​only remain unchanged or oscillate at the points where the sine takes the values ​​−1 and 1. However, since the number of points at which the sine takes these values ​​is only countably infinite and countably infinite sets have the Lebesgue measure 0, the exception set required in the definition from the point-wise convergence can be defined as ${\displaystyle (-1,1)}$ ${\displaystyle n}$

${\displaystyle A=\{x\in \mathbb {R} \,|\,\sin(x)=\pm 1\}}$.

Outside this set, i.e. on , there is point-wise convergence, the set has the Lebesgue measure 0, so the sequence of functions converges point-wise λ-almost everywhere to 0. ${\displaystyle \mathbb {R} \setminus A}$

Relationship to other convergence terms

Almost uniform convergence

From the almost uniform convergence follows the point-wise convergence μ-almost everywhere. Because by definition there is a set for every null sequence such that and that converges on uniformly. But then there is a null set and the sequence of functions converges point by point to the function ${\displaystyle \delta _{k}}$ ${\displaystyle A_{k}}$ ${\displaystyle \mu (A_{k})<\delta _{k}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X\setminus A_{k}}$ ${\displaystyle A=\cap _{k=1}^{\infty }A_{k}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$

${\displaystyle f(x)=\lim _{n\to \infty }f_{n}(x)\chi _{X\setminus A}(x)}$

and thus almost everywhere.

In the case of a finite measurement space , Jegorow's theorem also provides the inverse, i.e. that the almost uniform convergence follows from the point-wise convergence μ-almost everywhere. Thus for finite dimensional spaces the point-wise convergence almost everywhere and the almost uniform convergence coincide. The following example shows that the conclusion from point-wise convergence almost everywhere to almost uniform convergence for non-finite measure spaces is generally wrong. Looking at the sequence of functions

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

on the measure space , this sequence of functions converges pointwise almost everywhere to 0, because for anything is forever ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),\lambda )}$ ${\displaystyle x}$ ${\displaystyle n\geq x+2}$

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

But the sequence does not converge almost uniformly to 0, because it is forever with and thus ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle 0\leq \mu (A)<1}$ ${\displaystyle [n,n+1]\setminus A\neq \emptyset }$

${\displaystyle \sup _{x\in \mathbb {R} \setminus A}|f_{n}(x)-0|=1}$

for everyone with . So there cannot be an almost uniform convergence. ${\displaystyle A}$ ${\displaystyle \mu (A)<1}$

Customized convergence

From the point-wise convergence μ-almost everywhere follows in the case of a finite measure space the convergence according to measure , since then the theorem of Jegorow applies and the almost uniform convergence implies the convergence according to measure.

The finiteness of the dimensional space can not be dispensed with here, as the following example shows: for the dimensional space is the sequence of functions ${\displaystyle (\mathbb {R} ,{\mathcal {B}},\lambda )}$

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

for all point-wise convergent to 0. But it is not convergent to 0 by measure, because for is . ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle \varepsilon \in (0,1]}$ ${\displaystyle \lim _{n\to \infty }\lambda (\{|\chi _{[n,n+1]}-0|\geq \varepsilon \})=1}$

The inversion, i.e. the conclusion from convergence to measure convergence almost everywhere, does not apply to finite measure spaces, as the example in the section Convergence locally to measure shows.

Customized local convergence

From the point-wise convergence μ-almost everywhere, the convergence follows locally to measure . Because limits to the measure space on a lot with a so considers the measure space . This restricted measure space is a finite measure space, so Jegorow's theorem applies there . This provides the almost uniform convergence on the restricted dimensional space, which in turn implies the convergence to measure. Since this conclusion holds for every restriction to sets of finite measure, the function sequence converges to locally according to measure. ${\displaystyle A}$ ${\displaystyle \mu (A)<\infty }$ ${\displaystyle (A,{\mathcal {A}}|_{A},\mu |_{A})}$ ${\displaystyle (X,{\mathcal {A}},\mu )}$

The convergence does not apply, however, so convergence does not follow from convergence locally according to measure, convergence almost everywhere. An example can be constructed as follows: Consider the intervals

${\displaystyle (I_{n})_{n\in \mathbb {N} }=[0,1],[0,{\tfrac {1}{2}}],[{\tfrac {1}{2}},1],[0,{\tfrac {1}{3}}],[{\tfrac {1}{3}},{\tfrac {2}{3}}],[{\tfrac {2}{3}},1],[0,{\tfrac {1}{4}}],[{\tfrac {1}{4}},{\tfrac {2}{4}}],\dots }$

Then the sequence of functions converges

${\displaystyle f_{n}(x)=\chi _{I_{n}}(x)}$

on the measure space locally according to measure towards 0, because for is . But the sequence of functions does not converge point-by-point almost everywhere to 0, because an arbitrary one is contained in an infinite number and is also not contained in an infinite number . Thus, at every point, the values ​​0 and 1 take on infinitely often, so it cannot converge. ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda |_{[0,1]})}$ ${\displaystyle \varepsilon \in (0,1]}$ ${\displaystyle \lim _{n\to \infty }\lambda (\{f_{n}\geq \varepsilon \})=\lim _{n\to \infty }\lambda (I_{n})=0}$ ${\displaystyle x}$ ${\displaystyle I_{n}}$ ${\displaystyle I_{n}}$ ${\displaystyle \chi _{I_{n}}}$

Convergence in the pth mean

The point-wise convergence μ-almost everywhere does not generally result in the convergence in the p-th mean . Likewise, the convergence in the p-th mean does not generally result in the point-wise convergence μ-almost everywhere.

An example of this is the sequence of functions

${\displaystyle f_{n}(x)=n^{2}\chi _{[0,{\tfrac {1}{n}}]}(x)}$.

on the Maßram . It will almost certainly converge pointwise to 0, but it is ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda )}$

${\displaystyle \|f_{n}\|_{1}=n\;{\text{ und damit }}\lim _{n\to \infty }\|f_{n}\|_{1}=\infty }$.

Conversely, if one considers the sequence of intervals

${\displaystyle (I_{n})_{n\in \mathbb {N} }=[0,1],[0,{\tfrac {1}{2}}],[{\tfrac {1}{2}},1],[0,{\tfrac {1}{3}}],[{\tfrac {1}{3}},{\tfrac {2}{3}}],[{\tfrac {2}{3}},1],[0,{\tfrac {1}{4}}],[{\tfrac {1}{4}},{\tfrac {2}{4}}],\dots }$

and defines the sequence of functions as

${\displaystyle f_{n}(x)=\chi _{I_{n}}(x)}$,

so because the width of the intervals converges to zero. However, the sequence does not almost certainly converge to 0 point by point, since each of the values ​​0 and 1 is assumed as often as desired at any point . ${\displaystyle \lim _{n\to \infty }\|f_{n}\|_{1}=0}$ ${\displaystyle x}$

However, every sequence convergent in the p-th mean has an almost certainly convergent subsequence with the same limit value. For example, in the example above, one could select indices so that ${\displaystyle n_{k}}$

${\displaystyle I_{n_{k}}=[0,{\tfrac {1}{m}}]}$

for is. Then they almost certainly converge point by point to 0. ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle f_{n_{k}}}$

One criterion under which the point-wise convergence μ-almost everywhere the convergence in the p-th mean follows, is provided by the theorem of majorized convergence . It says that if in addition to convergence there is a majorant from almost everywhere , the convergence in the p-th mean also follows. More generally, it is sufficient if, instead of the existence of a majorant, only the uniform integrability of the sequence of functions is required, because from the convergence almost everywhere the convergence follows locally according to measure. Thus, with the same degree of integrability in the p-th mean, the convergence in the p-th mean can be deduced by means of Vitali's convergence theorem. From this perspective, the majorante is merely a sufficient criterion for equal integrability. ${\displaystyle {\mathcal {L}}^{p}}$

General formulation

The convergence almost everywhere can be defined analogously for images in more general image spaces, for example in topological spaces or in metric spaces . It should be noted here that the amount

${\displaystyle M:=\{x\in X\;|\;\lim _{n\to \infty }f_{n}(x)\neq f(x)\}}$

of the arguments, for which the function sequence does not converge point by point, does not have to be a measurable set , i.e. it may not be an element of . It is only required that a (measurable) null set exists with , and that the function sequence converges point by point. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle N}$ ${\displaystyle N\supset M}$ ${\displaystyle X\setminus N}$

Usually the convergence is defined almost everywhere for mappings with values ​​in a separable metric space, provided with Borel's σ-algebra . Then all sets are of the form

${\displaystyle \{x\in X\;|\;d(f(x),g(x))\geq \epsilon \}}$

measurable, i.e. included. Assigning a measure to quantities of this form allows certain alternative characterizations of the convergence. ${\displaystyle {\mathcal {A}}}$

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 .