Σ-finite measure

The term - finiteness (even - finiteness ) is used in the mathematical measure theory and provides a gradation of ( measurable ) sets of infinite measure in - finite and non- finite sets. It is introduced for reasons similar to the concept of countability with respect to the number of elements of a set.

Definition

A positive measure defined on an algebra over a base set is called - finally, when it countably many measurable sets of finite measure, that is, whose union is. The measure space is then also - finally called. A lot for which the measure space is limited to this - finally, ie - finite set.

The definition can be extended to signed Dimensions: A signed measure is called finite - if - finite.

Application

• Non- finite measures may have pathological properties, but many of the frequently observed dimensions are not finite. The class of finite - extent shares with the finite dimensions of some pleasant properties, finiteness can be compared in this respect with the separability of topological spaces. Some sets of analysis, such as the Radon - Nikodym and Fubini's theorem apply, for example, no longer for non- finite measures (but sometimes is a transfer to more general cases are possible by applying the rate for all - finite subspaces ).
• The Birkhoff integral for Banach space - valued functions is defined by means of finite - moderation.

Examples

• The Lebesgue measure on the real numbers is not finite, but infinite. After considering the intervals for all integers, so each interval has the dimension 1, and their association.
• This is a locally compact topological group - compact, so it's Hair measure - finite.

Content and Prämaße

Entirely analogously, one also speaks on semirings of finite - contents and Prämaßen. After Maßerweiterungssatz of Carathéodory each finite - premeasure is on a half-ring uniquely to a measure on the generated algebra resumable (without- finiteness does not follow the uniqueness).