A measure is a nonnegative, countably additive function defined on a collection of sets, usually a sigma-algebra. The precise definitions of measure and measurability provide a rigorous basis for integration. These definitions also underlie much of analysis and 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.
More generally, a measure 
on a delta-ring 
takes values in ![[0,infty]](/images/equations/Measure/Inline3.svg)
and satisfies
 |
(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 the underlying space is a countable union of sets in 
of finite measure,
then 
can be extended uniquely to a measure
defined on the sigma-algebra generated by 
.
is said to be a probability
measure on a set 
if 
and 
is a sigma-algebra.
In a special set-theoretic convention, a nontrivial countably additive measure is a probability
measure 
on the full power set 
of an infinite set 
that vanishes on every singleton
set (Jech 1997). This convention imposes additional requirements beyond those
of an ordinary probability measure, which
need only be defined on a sigma-algebra and may
assign positive measure to singleton
sets. The special convention requires:
1. 
and 
,
2. If 
then 
,
3. 
for all 
(nontriviality),
4. If 
, 
, 1, 2, ..., are pairwise disjoint, then
 |
(3)
|
Such a measure cannot exist on a countably infinite set, since countable additivity
would give 
.
A measure 
may be extended by completion. All subsets
of sets in 
with measure zero form a delta-ring 
. By "changing" sets
in 
on a set from
, a delta-ring 
which is the completion of 
with respect to 
is obtained.
The measure 
is called a complete measure
if 
. If 
is not complete, it may be extended to 
by setting 
, where 
and 
.
