A measure space is a basic object of
measure theory, a branch of
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 ...
that studies generalized notions of
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
s. It contains an underlying set, 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 of this set that are feasible for measuring (the
-algebra) and the method that is used for measuring (the
measure). One important example of a measure space is a
probability space.
A
measurable space consists of the first two components without a specific measure.
Definition
A measure space is a triple
where
*
is a set
*
is a
-algebra on the set
*
is a
measure on
In other words, a measure space consists of a
measurable space together with a
measure on it.
Example
Set
. The
-algebra on finite sets such as the one above is usually the
power set, which is the set of all subsets (of a given set) and is denoted by
Sticking with this convention, we set
In this simple case, the power set can be written down explicitly:
As the measure, define
by
so
(by additivity of measures) and
(by definition of measures).
This leads to the measure space
It is a
probability space, since
The measure
corresponds to the
Bernoulli distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabi ...
with
which is for example used to model a fair coin flip.
Important classes of measure spaces
Most important classes of measure spaces are defined by the properties of their associated measures. This includes
*
Probability spaces, a measure space where the measure is a
probability measure
* Finite measure spaces, where the measure is a
finite measure
*
-finite measure spaces, where the measure is a
-finite measure
Another class of measure spaces are the
complete measure space
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (''X'', Σ, ''μ'') is comp ...
s.
References
{{Measure theory
Measure theory