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.
