In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 a ...
that is, in some sense, a precursor to a ''
bona fide
In human interactions, good faith ( la, bona fides) is a sincere intention to be fair, open, and honest, regardless of the outcome of the interaction. Some Latin phrases have lost their literal meaning over centuries, but that is not the case ...
''
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
and
relative complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is th ...
) of a fixed set
and let