A measurable function 
on a measurable space 
is a function
for which, for every real number 
, the set
 |
is measurable (Hunter 2011, pp. 34-35).
More generally, a measurable function 
between measurable
spaces is a function for which the pre-image 
belongs to 
for every 
(Hunter 2011, p. 33).
Real-valued 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 real-valued 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 (Hunter 2011, pp. 50-51).
