Difference between revisions of "Approximate continuity"
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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 | 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 $f(x_0)$. It follows from [[Luzin-C-property|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). | + | [[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]]). The almost everywhere approximate continuity becomes then a characterization of measurability (Stepanov–Denjoy theorem). |
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 | 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 | ||
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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\, , | \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. | + | 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==== | ====References==== | ||
{| | {| | ||
| − | |valign="top"|{{Ref|Br}}|| A.M. Bruckner, "Differentiation of real functions" , Springer (1978) | + | |valign="top"|{{Ref|Br}}|| A.M. Bruckner, "Differentiation of real functions" , Springer (1978) {{MR|0507448}} {{ZBL|0382.26002}} |
|- | |- | ||
| − | |valign="top"|{{Ref| | + | |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| | + | |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| | + | |valign="top"|{{Ref|Mu}}|| M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953) {{MR|035237}} {{ZBL|0227.28001}} |
|- | |- | ||
| − | |valign="top"|{{Ref|Th}}|| B.S. Thomson, "Real functions" , Springer (1985) | + | |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}} | ||
|- | |- | ||
|} | |} | ||
Revision as of 07:49, 16 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 $f(x_0)$ (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). 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 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
| [Br] | A.M. Bruckner, "Differentiation of real functions" , Springer (1978) MR0507448 Zbl 0382.26002 |
| [EG] | L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800 |
| [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 |
| [Sa] | S. Saks, "Theory of the integral" , Hafner (1952) MR0167578 Zbl 63.0183.05 |
| [Th] | B.S. Thomson, "Real functions" , Springer (1985) MR0818744 Zbl 0581.26001 |
Approximate continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_continuity&oldid=27431