Approximate continuity

2020 Mathematics Subject Classification: Primary: 26B05 Secondary: 28A20 [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 $f(x_0)$ (cp. with Section 1.7.2 of [EG]). It follows from Lusin's theorem that a measurable function is approximately continuous at almost every point (see Theorem 3 of Section 1.7.2 of [EG]). The definition of approximate continuity can be extended to nonmeasurable functions (cp. with Approximate limit and see Section 2.9.12 of [Fe]). The almost everywhere approximate continuity becomes then a characterization of measurability (Stepanov–Denjoy theorem, see Theorem 2.9.13 of [Fe]).

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 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 [EG]). Conversely, if $f$ is essentially bounded, the points of approximate continuity of $f$ are also Lebesgue points.

References

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[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Mu]	M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953) MR035237 Zbl 0227.28001
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