In mathematics, a decomposable measure (also known as a strictly localizable measure) is a
measure that is a
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...
of
finite measure
In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than ...
s. This is a generalization of
σ-finite measures, which are the same as those that are a disjoint union of ''
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
'' finite measures. There are several theorems in
measure theory such as the
Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measu ...
that are not true for arbitrary measures but are true for σ-finite measures. Several such theorems remain true for the more general class of decomposable measures. This extra generality is not used much as most decomposable measures that occur in practice are σ-finite.
Examples
*
Counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infin ...
on an uncountable measure space with all subsets measurable is a decomposable measure that is not σ-finite.
Fubini's theorem and Tonelli's theorem hold for σ-finite measures but can fail for this measure.
* Counting measure on an uncountable measure space with not all subsets measurable is generally not a decomposable measure.
* The one-point space of measure infinity is not decomposable.
References
Bibliography
*
* Second printing.
{{Measure theory
Measures (measure theory)