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".
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Classical statement
For an interval [a, b], let
be a measurable function. Then, for every ε > 0, there exists a compact E ⊂ [a, b] such that f restricted to E is continuous and
Note that E inherits the subspace topology from [a, b]; continuity of f restricted to E is defined using this topology.
General form
Let
be a measurable function. Given ε > 0, for every
Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.