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Lusin's Theorem


Let f(x) be a finite and measurable function in (-infty,infty), and let epsilon be freely chosen. Then there is a function g(x) such that

1. g(x) is continuous in (-infty,infty),

2. The measure of {x:f(x)!=g(x)} is <epsilon,

3. M(|g|;R)<=M(|f|;R),

where M(f;S) denotes the upper bound of the set of the values of f(P) as P runs through all values of S.


See also

Measurable Function, Measure

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References

Kestelman, H. §4.4 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 30 and 109-112, 1960.

Referenced on Wolfram|Alpha

Lusin's Theorem

Cite this as:

Weisstein, Eric W. "Lusin's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LusinsTheorem.html

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