Lusin's theorem

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In the mathematical field of real analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) states that every measurable function is a continuous function on nearly all its domain. In the informal formulation of J. E. Littlewood, "every measurable function is nearly continuous".

Classical statement[edit]

For an interval [ab], let

be a measurable function. Then, for every ε > 0, there exists a compact E ⊂ [ab] such that f restricted to E is continuous and

Note that E inherits the subspace topology from [ab]; continuity of f restricted to E is defined using this topology.

General form[edit]

Let be a Radon measure space and Y be a second-countable topological space, let

be a measurable function. Given ε > 0, for every of finite measure there is a closed set E with µ(A \ E) < ε such that f restricted to E is continuous. If A is locally compact, we can choose E to be compact and even find a continuous function with compact support that coincides with f on E and such that .

Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.

References[edit]

  • N. Lusin. Sur les propriétés des fonctions mesurables, Comptes Rendus Acad. Sci. Paris 154 (1912), 1688–1690.
  • G. Folland. Real Analysis: Modern Techniques and Their Applications, 2nd ed. Chapter 7
  • W. Zygmunt. Scorza-Dragoni property (in Polish), UMCS, Lublin, 1990
  • M. B. Feldman, "A Proof of Lusin's Theorem", American Math. Monthly, 88 (1981), 191-2