A measurable function 
is a function for which, for every real number 
, the set
 |
is measurable.
Measurable functions are closed under addition and multiplication. The composition of measurable functions is measurable whenever the measurable-space structures on the intermediate domain and codomain agree.
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 sigma-algebra
of measurable sets in 
, not on the values of a particular measure.
Sometimes a standard sigma-algebra is understood.
For instance, a measurable function on 
is usually taken to be measurable with respect to the Lebesgue
sigma-algebra.
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.
