A function is measurable if, for every real number , the set
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 is measurable depends on the measure on , and, in particular, it only depends on the sigma-algebra of measurable sets in . Sometimes, the measure on may be assumed to be a standard measure. For instance, a measurable function on 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 where they differ has measure zero.