Namespaces
Variants
Actions

Difference between revisions of "Approximate limit"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 1: Line 1:
A limit of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128701.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128702.png" /> over a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128703.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128704.png" /> is a [[Density point|density point]]. In the simplest case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128705.png" /> is a real-valued function of the points of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128706.png" />-dimensional Euclidean space; in the more general case it is a vector function. The approximate limit is denoted by
+
{{MSC|28A33|49Q15}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128707.png" /></td> </tr></table>
+
[[Category:Classical measure theory]]
  
In general, the existence of an ordinary limit does not follow from the existence of an approximate limit. An approximate limit displays the elementary properties of limits — uniqueness, and theorems on the limit of a sum, difference, product and quotient of two functions.
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128708.png" /> be a density point of the domain of definition of a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a0128709.png" />. If the ordinary limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287010.png" /> exists, the approximate limit also exists and is equal to it. The approximate upper limit of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287011.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287012.png" /> is the lower bound of the set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287013.png" /> (including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287014.png" />) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287015.png" /> is a point of dispersion of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287016.png" />. Similarly, the approximate lower limit of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287017.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287018.png" /> is the upper bound of the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287019.png" /> (including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287020.png" />) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287021.png" /> is a point of dispersion of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287022.png" />. These approximate limits are denoted, respectively, by
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287023.png" /></td> </tr></table>
+
A concept of classical measure theory.
  
An approximate limit exists if and only if the approximate upper and lower limits are equal; their common value is equal to the approximate limit.
+
====Definition====
 +
Consider a (Lebesgue)-measurable set $E\subset \mathbb R^n$, a measurable function $f: E\to \mathbb R$ and a point $x_0\in \mathbb R^n$ where $E$ has Lebesgue density $1$ (see [[Density of a set]]). The approximate upper and lower limits of $f$ at $x_0$ are defined, respectively, as
 +
* The infimum of $a\in \mathbb R\cup\{\infty\}$ such that the set $\{f\leq a\}$ has density $1$ at $x_0$;
 +
* The supremum of $a\in\{-\infty\}\cup\mathbb R$ such that the set $\{f\geq a\}$ has density $1$ at $x_0$.
 +
They are usually denoted by
 +
\[
 +
{\rm ap}\,\limsup_{x\to x_0}\, f(x) \qquad \mbox{and}\qquad {\rm ap}\, \liminf_{x\to x_0}\, f(x)
 +
\]
 +
(some authors use also the notation $\overline{\lim}\,{\rm ap}$ and $\underline{\lim}\,{\rm ap}$). It follows from the definition that ${\rm ap}\, \liminf\leq {\rm}\, {\rm ap}\,\limsup$: if the two numbers coincide then the result is called ''approximate limit of $f$ at $x_0$'' and it is denoted by
 +
\[
 +
{\rm ap}\,\lim_{x\to x_0}\, f(x)\, .
 +
\]
 +
The approximate limit of a function taking values in a finite-dimensional vector space can be defined using its coordinate functions and the definition above.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287024.png" /> is real, one-sided (right and left) approximate upper and lower limits are also used (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287025.png" /> must then be, respectively, a right-hand or left-hand density point in the domain of definition of the function). For the approximate right upper limit the following notation is used:
+
====Properties====
 +
Observe that the approximate limit of $f$ and $g$ are the same if $f$ and $g$ differ on a set of measure zero. A useful characterization of the approximate limit is given by the following
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287026.png" /></td> </tr></table>
+
'''Proposition 1'''
 +
Consider a (Lebesgue)-measurable set $E\subset \mathbb R^n$, a  measurable function $f: E\to \mathbb R$ and a point $x_0\in \mathbb R^n$. $f$ has approximate limit $L$ at $x_0$ if and only if there is a measurable set $F\subset E$ which has density $1$ at $x_0$ and such that
 +
\[
 +
\lim_{x\in F, x\to x_0} f(x) = L\, .
 +
\]
 +
In general, the existence of an ordinary limit does not follow from the  existence of an approximate limit. An approximate limit displays the  elementary properties of limits — uniqueness, and theorems on the limit  of a sum, difference, product and quotient of two functions — these properties follow indeed easily from Proposition 1.
  
with corresponding notations for the other cases. If the approximate right upper and lower limits coincide, one obtains the right approximate limit; if the approximate left upper and lower limits coincide, one obtains the left approximate limit.
+
====One-sided approximate limits====
 +
If the domain $E$ of $f$ is a subset of $\mathbb R$ we can define  one-sided (right and left) approximate upper and lower limits: we just substitute all ''density $1$'' requirements with the ''right-hand'' or the ''left-hand density $1$'' requirement, that are, respectively,
 +
\[
 +
\lim_{r\downarrow 0} \frac{\lambda (E\cap ]x_0, x_0+r[)}{r} = 1 \qquad \mbox{and}\qquad \lim_{r\downarrow 0}  \frac{\lambda (E\cap ]x_0-r, x_0[)}{r} = 1\,
 +
\]
 +
(here $\lambda$ denotes the Lebesgue measure on $\mathbb R$). The notation then becomes, for instance,
 +
\[
 +
{\rm ap}\, \lim_{x\to x_0^+} f(x)
 +
\]
 +
for the right approximate limit. An analogous notation is used for all the other objects.
  
Approximate limits were first utilized by A. Denjoy and A.Ya. Khinchin in the study of the differential connections between an indefinite integral (in the sense of Lebesgue and in the sense of Denjoy–Khinchin) and the integrand (cf. [[Approximate continuity|Approximate continuity]]; [[Approximate derivative|Approximate derivative]]).
+
 +
====Comments====
 +
Approximate limits are used to define [[Approximate continuity|approximately continuous]] and [[Approximate differentiability|approximate differentiable]] functions.
  
====References====
+
Approximate limits were first utilized by A. Denjoy and A.Ya. Khinchin in the study of the differential connections between an indefinite integral (in the sense of Lebesgue and in the sense of Denjoy–Khinchin).
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner  (1952) (Translated from French)</TD></TR></table>
 
  
 
+
The definitions are sometimes extended to non-measurable functions: in that case the Lebesgue measure is substituted by the Lebesgue [[Outer measure|outer measure]] (cp. with [[Density of a set]]).
 
 
====Comments====
 
A point of dispersion is defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287027.png" /> be a set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287029.png" /> be the completely-additive set function defined for measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287030.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287031.png" />, the outer measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287032.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287033.png" /> be any point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287034.png" />. The upper strong derivative and lower strong derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287036.png" /> are called, respectively, the upper outer density and lower outer density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287037.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287038.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287039.png" /> is a point of density for a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287040.png" /> if the outer density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287041.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287042.png" /> is 1 and it is a point of dispersion if the outer density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287043.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287044.png" /> is zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287045.png" /> is measurable, almost-all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287046.png" /> are points of density and almost-all points of its complement are points of dispersion. The latter condition is also sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012870/a01287047.png" /> to be measurable.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Bruckner,  "Differentiation of real functions" , Springer  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M.E. Munroe,  "Introduction to measure and integration" , Addison-Wesley  (1953) pp. 111</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"B.S. Thomson,  "Real functions" , Springer  (1985)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|Br}}|| A.M. Bruckner,  "Differentiation of real functions" , Springer  (1978)
 +
|-
 +
|valign="top"|{{Ref|Fe}}||  H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren  der mathematischen Wissenschaften. Springer-Verlag New York Inc., New  York, 1969.
 +
|-
 +
|valign="top"|{{Ref|Mu}}|| M.E. Munroe,  "Introduction to measure and integration" , Addison-Wesley  (1953)
 +
|-
 +
|valign="top"|{{Ref|Sa}}|| S. Saks,  "Theory of the integral" , Hafner  (1952)
 +
|-
 +
|valign="top"|{{Ref|Th}}|| B.S. Thomson,  "Real functions" , Springer  (1985)
 +
|-
 +
|}

Revision as of 06:27, 6 August 2012

2020 Mathematics Subject Classification: Primary: 28A33 Secondary: 49Q15 [MSN][ZBL]

A concept of classical measure theory.

Definition

Consider a (Lebesgue)-measurable set $E\subset \mathbb R^n$, a measurable function $f: E\to \mathbb R$ and a point $x_0\in \mathbb R^n$ where $E$ has Lebesgue density $1$ (see Density of a set). The approximate upper and lower limits of $f$ at $x_0$ are defined, respectively, as

  • The infimum of $a\in \mathbb R\cup\{\infty\}$ such that the set $\{f\leq a\}$ has density $1$ at $x_0$;
  • The supremum of $a\in\{-\infty\}\cup\mathbb R$ such that the set $\{f\geq a\}$ has density $1$ at $x_0$.

They are usually denoted by \[ {\rm ap}\,\limsup_{x\to x_0}\, f(x) \qquad \mbox{and}\qquad {\rm ap}\, \liminf_{x\to x_0}\, f(x) \] (some authors use also the notation $\overline{\lim}\,{\rm ap}$ and $\underline{\lim}\,{\rm ap}$). It follows from the definition that ${\rm ap}\, \liminf\leq {\rm}\, {\rm ap}\,\limsup$: if the two numbers coincide then the result is called approximate limit of $f$ at $x_0$ and it is denoted by \[ {\rm ap}\,\lim_{x\to x_0}\, f(x)\, . \] The approximate limit of a function taking values in a finite-dimensional vector space can be defined using its coordinate functions and the definition above.

Properties

Observe that the approximate limit of $f$ and $g$ are the same if $f$ and $g$ differ on a set of measure zero. A useful characterization of the approximate limit is given by the following

Proposition 1 Consider a (Lebesgue)-measurable set $E\subset \mathbb R^n$, a measurable function $f: E\to \mathbb R$ and a point $x_0\in \mathbb R^n$. $f$ has approximate limit $L$ at $x_0$ if and only if there is a measurable set $F\subset E$ which has density $1$ at $x_0$ and such that \[ \lim_{x\in F, x\to x_0} f(x) = L\, . \] In general, the existence of an ordinary limit does not follow from the existence of an approximate limit. An approximate limit displays the elementary properties of limits — uniqueness, and theorems on the limit of a sum, difference, product and quotient of two functions — these properties follow indeed easily from Proposition 1.

One-sided approximate limits

If the domain $E$ of $f$ is a subset of $\mathbb R$ we can define one-sided (right and left) approximate upper and lower limits: we just substitute all density $1$ requirements with the right-hand or the left-hand density $1$ requirement, that are, respectively, \[ \lim_{r\downarrow 0} \frac{\lambda (E\cap ]x_0, x_0+r[)}{r} = 1 \qquad \mbox{and}\qquad \lim_{r\downarrow 0} \frac{\lambda (E\cap ]x_0-r, x_0[)}{r} = 1\, \] (here $\lambda$ denotes the Lebesgue measure on $\mathbb R$). The notation then becomes, for instance, \[ {\rm ap}\, \lim_{x\to x_0^+} f(x) \] for the right approximate limit. An analogous notation is used for all the other objects.


Comments

Approximate limits are used to define approximately continuous and approximate differentiable functions.

Approximate limits were first utilized by A. Denjoy and A.Ya. Khinchin in the study of the differential connections between an indefinite integral (in the sense of Lebesgue and in the sense of Denjoy–Khinchin).

The definitions are sometimes extended to non-measurable functions: in that case the Lebesgue measure is substituted by the Lebesgue outer measure (cp. with Density of a set).

References

[Br] A.M. Bruckner, "Differentiation of real functions" , Springer (1978)
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969.
[Mu] M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953)
[Sa] S. Saks, "Theory of the integral" , Hafner (1952)
[Th] B.S. Thomson, "Real functions" , Springer (1985)
How to Cite This Entry:
Approximate limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_limit&oldid=27389
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article