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 measurable space or Borel space
is a basic object in
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 mass and probability of events. These seemingly distinct concepts have many simila ...
. It consists of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
and a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
, which defines the
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s that will be measured.
Definition
Consider a set
and a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
on
Then the tuple
is called a measurable space.
Note that in contrast to a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
, no
measure is needed for a measurable space.
Example
Look at the set:
One possible
-algebra would be:
Then
is a measurable space. Another possible
-algebra would be the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
on
:
With this, a second measurable space on the set
is given by
Common measurable spaces
If
is finite or countably infinite, the
-algebra is most often the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
on
so
This leads to the measurable space
If
is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
, the
-algebra is most commonly the
Borel -algebra so
This leads to the measurable space
that is common for all topological spaces such as the real numbers
Ambiguity with Borel spaces
The term Borel space is used for different types of measurable spaces. It can refer to
* any measurable space, so it is a synonym for a measurable space as defined above
* a measurable space that is
Borel isomorphic to a measurable subset of the real numbers (again with the Borel
-algebra)
See also
*
*
*
References
{{Measure theory
Measure theory