The terms "measure," "measurable," etc. have very precise technical definitions (usually involving sigma-algebras) that can make them appear difficult to understand. However, the technical nature of the definitions is extremely important, since it gives a firm footing to concepts that are the basis for much of analysis (including some of the slippery underpinnings of calculus).
For example, every definition of an integral is based on a particular measure: the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The study of measures and their application to integration is known as measure theory.
A measure is defined as a nonnegative real function from a delta-ring 
such that
 |
(1)
|
where 
is the empty
set, and
 |
(2)
|
for any finite or countable collection of pairwise disjoint sets 
in 
such that 
is also in 
.
If 
is 
-finite and 
is bounded, then 
can be extended uniquely to a measure defined on the 
-algebra generated by 
.
is said to be a probability
measure on a set 
if 
and 
is a 
-algebra.
In the usual definition of a probability measure (or, more precisely a nontrivial 
-additive measure), a measure is a real-valued function
on the Power
Set 
of an infinite set 
that satisfies the following properties:
1. 
and 
,
2. If 
then 
,
3. 
for all 
(nontriviality),
4. If 
, are pairwise disjoint,
then
 |
(3)
|
(Jech 1997).
A measure 
may be extended by completion. The subsets of sets with measure zero form a 
-ring 
. By "changing" sets in 
on a set from 
, a 
-ring
which is the completion of 
with respect to 
is obtained.
The measure 
is called complete if 
.
If 
is not complete, it may be extended
to 
by setting 
, where 
and 
.
