Stepanov-Denjoy theorem

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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 E ⊆ R n {\displaystyle E\subseteq \mathbb {R} ^{n}} {\displaystyle E\subseteq \mathbb {R} ^{n}} be a Lebesgue measurable set, f : E → R k {\displaystyle f\colon E\to \mathbb {R} ^{k}} {\displaystyle f\colon E\to \mathbb {R} ^{k}} be a measurable function, and x 0 ∈ E {\displaystyle x_{0}\in E} {\displaystyle x_{0}\in E} be a point where the Lebesgue density of E {\displaystyle E} {\displaystyle E} is 1. The function f {\displaystyle f} {\displaystyle f} is said to be approximately continuous at x 0 {\displaystyle x_{0}} {\displaystyle x_{0}} if and only if the approximate limit of f {\displaystyle f} {\displaystyle f} at x 0 {\displaystyle x_{0}} {\displaystyle x_{0}} exists and equals f ( x 0 ) {\displaystyle f(x_{0})} {\displaystyle f(x_{0})}.[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 f ∈ L 1 ( E ) {\displaystyle f\in L^{1}(E)} {\displaystyle f\in L^{1}(E)}, a point x 0 {\displaystyle x_{0}} {\displaystyle x_{0}} is a Lebesgue point if it is a point of Lebesgue density 1 for E {\displaystyle E} {\displaystyle E} and satisfies

lim r ↓ 0 1 λ ( B r ( x 0 ) ) ∫ E ∩ B r ( x 0 ) | f ( x ) − f ( x 0 ) | d x = 0 {\displaystyle \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} {\displaystyle \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 λ {\displaystyle \lambda } {\displaystyle \lambda } denotes the Lebesgue measure and B r ( x 0 ) {\displaystyle B_{r}(x_{0})} {\displaystyle B_{r}(x_{0})} represents the ball of radius r {\displaystyle r} {\displaystyle r} centered at x 0 {\displaystyle x_{0}} {\displaystyle x_{0}}. Every Lebesgue point of a function is necessarily a point of approximate continuity.[6] The converse relationship holds under additional constraints: when f {\displaystyle f} {\displaystyle f} is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.[7]

See also

References

  1. "Approximate continuity". Encyclopedia of Mathematics. Retrieved January 7, 2025.
  2. Evans, L.C.; Gariepy, R.F. (1992). Measure theory and fine properties of functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press.
  3. Federer, H. (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. New York: Springer-Verlag.
  4. Saks, S. (1952). Theory of the integral. Hafner.
  5. Lukeš, Jaroslav (1978). "A topological proof of Denjoy-Stepanoff theorem". Časopis pro pěstování matematiky. 103 (1): 95–96. doi:10.21136/CPM.1978.117963. hdl:10338.dmlcz/117963. ISSN 0528-2195. Retrieved 2025-01-20.
  6. Thomson, B.S. (1985). Real functions. Springer.
  7. Munroe, M.E. (1953). Introduction to measure and integration. Addison-Wesley.