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Lusin's theorem
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==
For an interval (), let
:
be a measurable function. Then, for every ''ε'' > 0, there exists a compact ''E'' ⊂ () such that ''f'' restricted to ''E'' is continuous and
:
Note that ''E'' inherits the subspace topology from (); continuity of ''f'' restricted to ''E'' is defined using this topology.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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