View source for Approximate continuity

{{MSC|28A33|49Q15}}

[[Category:Classical measure theory]]


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A concept of classical measure theory.

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
[[Approximate limit|approximate limit]] of $f$ at $x_0$ exists and equals $f(x_0)$ (cp. with Section 1.7.2 of {{Cite|EG}}). It follows from [[Luzin-C-property|Lusin's theorem]] that a measurable function is approximately continuous at almost every point (see Theorem 3 of Section 1.7.2 of {{Cite|EG}}). The definition of approximate continuity can be extended to nonmeasurable functions (cp. with [[Approximate limit]] and see Section 2.9.12 of {{Cite|Fe}}). The almost everywhere approximate continuity becomes then a characterization of measurability (Stepanov–Denjoy theorem, see Theorem 2.9.13 of {{Cite|Fe}}).

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 of Lebesgue density $1$ for $E$ 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. In particular a Lebesgue point is always a point of approximate continuity
(cp. with Section 1.7.2 of {{Cite|EG}}). Conversely, if $f$ is essentially bounded, the points of approximate continuity of $f$ are also Lebesgue points. 

====References====
{|
|valign="top"|{{Ref|Br}}|| A.M. Bruckner,   "Differentiation of real functions" , Springer  (1978) {{MR|0507448}}  {{ZBL|0382.26002}}
|-
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy,  "Measure theory  and fine properties of functions" Studies in Advanced  Mathematics. CRC  Press, Boca Raton, FL,  1992. {{MR|1158660}}  {{ZBL|0804.2800}} 
|-
|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. {{MR|0257325}} {{ZBL|0874.49001}}
|-
|valign="top"|{{Ref|Mu}}||  M.E. Munroe,   "Introduction to measure and integration" , Addison-Wesley  (1953) {{MR|035237}} {{ZBL|0227.28001}} 
|-
|valign="top"|{{Ref|Sa}}|| S. Saks,   "Theory of the integral" , Hafner  (1952) {{MR|0167578}} {{ZBL|63.0183.05}}
|-
|valign="top"|{{Ref|Th}}|| B.S. Thomson,   "Real functions" , Springer  (1985) {{MR|0818744}} {{ZBL|0581.26001}} 
|-
|}