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 ...
, in the field of
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, Ï„-additivity is a certain property of
measures on
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s.
A measure or
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 ...
on a space
whose domain is a
sigma-algebra is said to be if for any
upward-directed family
of nonempty
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s such that its
union is in
the measure of the union is the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
of measures of elements of
that is,:
See also
*
*
*
References
* .
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Measure theory