In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a pre-measure is a
set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R ...
that is, in some sense, a precursor to a ''
bona fide''
measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.
Definition
Let
be a
ring of subsets (closed under
union and
relative complement
In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in .
When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set , the absolute complement ...
) of a fixed set
and let