HOME

TheInfoList



Click Here for Items Related To - Decomposable Measure
OR:
Decomposable Measure on:   [Wikipedia]   [Google]   [Amazon]

In mathematics, a decomposable measure (also known as a strictly localizable measure) is a measure that is a
disjoint union In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
of finite measures. This is a generalization of
σ-finite measure In mathematics, given a positive or a signed measure \mu on a measurable space (X, \mathcal F), a \sigma-finite subset is a measurable subset which is the union of a countable number of measurable subsets of finite measure. The measure \mu is ca ...
s, which are the same as those that are a disjoint union of '' countably many'' finite measures. There are several theorems 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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
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 measurab ...
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 infinit ...
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)