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A generalization of the concept of continuity in which the ordinary limit is replaced by an [[Approximate limit|approximate limit]]. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128401.png" /> is called approximately continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128402.png" /> if
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{{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/a012840/a0128403.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
In the simplest case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128404.png" /> is a real-valued function of the points of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128405.png" />-dimensional Euclidean space (in general it is a vector-valued function). The following theorems apply. 1) A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128406.png" /> is Lebesgue-measurable on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128407.png" /> if and only if it is approximately continuous almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128408.png" /> (the Stepanov–Denjoy theorem). 2) For any bounded Lebesgue-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a0128409.png" /> one has, at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a01284010.png" />,
 
  
<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/a012840/a01284011.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a01284012.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a01284013.png" />-dimensional Lebesgue measure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a01284014.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a01284015.png" />-dimensional non-degenerate segment containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a01284016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012840/a01284017.png" /> is its diameter.
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A concept of classical measure theory.
  
====References====
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A generalization of the concept of continuity in which the ordinary limit is replaced by an [[Approximate limit|approximate limit]]. Consider a (Lebesgue) measurable set $E\subset \mathbb R^n$, a measurable function $f: E\to \mathbb R^k$ and a point $x_0\in E$ where the [[Density of a set|Lebesgue density]] of $E$ is $1$. $f$ is approximately continuous at $x_0$ if and only if the
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952) (Translated from French)</TD></TR></table>
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[[Approximate limit|approximate limit]] of $f$ at $x_0$ exists and equals $f(x_0)$. It follows from [[Luzin theorem|Lusin's theorem]] that a measurable function is approximately continuous at almost every point. The definition of approximate continuity can be extended to nonmeasurable functions (cp. with [[Approximate limit]]). The almost everywhere approximate continuity becomes then a characterization of measurability (Stepanov–Denjoy theorem).
  
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Points of approximate continuity are related to [[Lebesgue point|Lebesgue points]]. Recall that a Lebesgue point $x_0$ of a function $f\in L^1 (E)$ is a point at which
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\[
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\lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x_0))} \int_{E\cap B_r (x_0)} |f(x)-f(x_0)|\, dx = 0\, ,
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\]
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where $\lambda$ denotes the Lebesgue measure. Thus, if $f$ is essentially bounded, the points of approximate continuity of $f$ are precisely its Lebesgue points.
  
 
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====References====
====Comments====
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{|
For other references see [[Approximate limit|Approximate limit]].
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|valign="top"|{{Ref|Br}}|| A.M. Bruckner,  "Differentiation of real functions" , Springer  (1978)
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|-
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|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.
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|-
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|valign="top"|{{Ref|Mu}}||  M.E. Munroe,  "Introduction to measure and integration" , Addison-Wesley  (1953)
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|-
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|valign="top"|{{Ref|Sa}}|| S. Saks,  "Theory of the integral" , Hafner  (1952)
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|-
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|valign="top"|{{Ref|Th}}|| B.S. Thomson,  "Real functions" , Springer  (1985)
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|-
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|}

Revision as of 06:41, 6 August 2012

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

A concept of classical measure theory.

A generalization of the concept of continuity in which the ordinary limit is replaced by an approximate limit. Consider a (Lebesgue) measurable set $E\subset \mathbb R^n$, a measurable function $f: E\to \mathbb R^k$ and a point $x_0\in E$ where the Lebesgue density of $E$ is $1$. $f$ is approximately continuous at $x_0$ if and only if the approximate limit of $f$ at $x_0$ exists and equals $f(x_0)$. It follows from Lusin's theorem that a measurable function is approximately continuous at almost every point. The definition of approximate continuity can be extended to nonmeasurable functions (cp. with Approximate limit). The almost everywhere approximate continuity becomes then a characterization of measurability (Stepanov–Denjoy theorem).

Points of approximate continuity are related to Lebesgue points. Recall that a Lebesgue point $x_0$ of a function $f\in L^1 (E)$ is a point at which \[ \lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x_0))} \int_{E\cap B_r (x_0)} |f(x)-f(x_0)|\, dx = 0\, , \] where $\lambda$ denotes the Lebesgue measure. Thus, if $f$ is essentially bounded, the points of approximate continuity of $f$ are precisely its Lebesgue points.

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 continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_continuity&oldid=27390
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article