Approximate continuity
This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.
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 ). 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 ). The definition of approximate continuity can be extended to nonmeasurable functions (cp. with Approximate limit and see Section 2.9.12 of ). The almost everywhere approximate continuity becomes then a characterization of measurability (Stepanov–Denjoy theorem, see Theorem 2.9.13 of ).
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 ). Conversely, if $f$ is essentially bounded, the points of approximate continuity of $f$ are also Lebesgue points.
References
| [1] | A.M. Bruckner, "Differentiation of real functions" , Springer (1978) MR0507448 Template:ZBL |
| [2] | L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Template:ZBL |
| [3] | H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Template:ZBL |
| [4] | M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953) MR035237 Template:ZBL |
| [5] | S. Saks, "Theory of the integral" , Hafner (1952) MR0167578 Template:ZBL |
| [6] | B.S. Thomson, "Real functions" , Springer (1985) MR0818744 Template:ZBL |
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