Approximately continuous function
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In mathematics, particularly in mathematical analysis and measure theory, an approximately continuous function is a concept that generalizes the notion of continuous functions by replacing the ordinary limit with an approximate limit.[1] This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory.[2]
Definition
Let be a Lebesgue measurable set, be a measurable function, and be a point where the Lebesgue density of is 1. The function is said to be approximately continuous at if and only if the approximate limit of at exists and equals .[3]
Properties
A fundamental result in the theory of approximately continuous functions is derived from Lusin's theorem, which states that every measurable function is approximately continuous at almost every point of its domain.[4] The concept of approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces. The Stepanov-Denjoy theorem provides a remarkable characterization:
Stepanov-Denjoy theorem: A function is measurable if and only if it is approximately continuous almost everywhere. [5]
Approximately continuous functions are intimately connected to Lebesgue points. For a function , a point is a Lebesgue point if it is a point of Lebesgue density 1 for and satisfies
where denotes the Lebesgue measure and represents the ball of radius centered at . Every Lebesgue point of a function is necessarily a point of approximate continuity.[6] The converse relationship holds under additional constraints: when is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.[7]
See also
- Approximate limit
- Density point
- Density topology (which serves to describe approximately continuous functions in a different way, as continuous functions for a different topology)
- Lebesgue point
- Lusin's theorem
- Measurable function
References
- ↑ "Approximate continuity". https://encyclopediaofmath.org/wiki/Approximate_continuity.
- ↑ Evans, L.C.; Gariepy, R.F. (1992). Measure theory and fine properties of functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press.
- ↑ Federer, H. (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. 153. New York: Springer-Verlag.
- ↑ Saks, S. (1952). Theory of the integral. Hafner.
- ↑ Lukeš, Jaroslav (1978). "A topological proof of Denjoy-Stepanoff theorem". Časopis pro pěstování matematiky 103 (1): 95–96. doi:10.21136/CPM.1978.117963. ISSN 0528-2195. https://dml.cz/handle/10338.dmlcz/117963. Retrieved 2025-01-20.
- ↑ Thomson, B.S. (1985). Real functions. Springer.
- ↑ Munroe, M.E. (1953). Introduction to measure and integration. Addison-Wesley.
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