Approximate continuity

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)