Supriya Ghosh (Editor)

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".

Contents

Classical statement

For an interval [a, b], let

f : [ a , b ] → C

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

μ ( E ) > b − a − ε .

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

General form

Let ( X , Σ , μ ) be a Radon measure space and Y be a second-countable topological space, let

f : X → Y

be a measurable function. Given ε > 0, for every A ∈ Σ 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 f ε : X → Y with compact support that coincides with f on E and such that   sup x ∈ X | f ε ( x ) | ≤ sup x ∈ X | f ( x ) | .

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

References

Lusin's theorem Wikipedia
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