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Measurable Function


A function f:X->R is measurable if, for every real number a, the set

 {x in X:f(x)>a}

is measurable.

Measurable functions are closed under addition and multiplication, but not composition.

The measurable functions form one of the most general classes of real functions. They are one of the basic objects of study in analysis, both because of their wide practical applicability and the aesthetic appeal of their generality. Whether a function f:X->R is measurable depends on the measure mu on X, and, in particular, it only depends on the sigma-algebra of measurable sets in X. Sometimes, the measure on X may be assumed to be a standard measure. For instance, a measurable function on R is usually measurable with respect to Lebesgue measure.

From the point of view of measure theory, subsets with measure zero do not matter. Often, instead of actual real-valued functions, equivalence classes of functions are used. Two functions are equivalent if the subset of the domain X where they differ has measure zero.


See also

Borel Measure, Lebesgue Measure, Measure, Measure Space, Measure Theory, Real Function, Sigma-Algebra

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Measurable Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MeasurableFunction.html

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