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...

measurable subsets of finite measure. The measure μ {\displaystyle \mu } is called a σ {\displaystyle \sigma } -finite measure if the set X {\displaystyle...

mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure. Let ( X , T ) {\displaystyle...

of finite measure. Analogously, a set in a measure space is said to have a σ-finite measure if it is a countable union of sets with finite measure. For...

In measure theory, a branch of mathematics that studies generalized notions of volumes, an s-finite measure is a special type of measure. An s-finite measure...

that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible"...

measure is a measure defined on infinite-dimensional normed vector spaces, such as Banach spaces, which resembles the Lebesgue measure used in finite-dimensional...

signed and complex measures: namely, that if μ {\displaystyle \mu } is a nonnegative σ-finite measure, and ν {\displaystyle \nu } is a finite-valued signed...

σ-finite measure can be decomposed into the sum of an absolutely continuous measure and a singular measure with respect to another σ-finite measure. See...

Probability spaces, a measure space where the measure is a probability measure Finite measure spaces, where the measure is a finite measure σ {\displaystyle...

of taking the limit of finite systems. A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency...

(S).} Notice that the result may be equal to +∞, unless μ is a finite measure. A finite linear combination of indicator functions ∑ k a k 1 S k {\displaystyle...

atomic class. If μ {\displaystyle \mu } is a σ {\displaystyle \sigma } -finite measure, there are countably many atomic classes. Consider the set X = {1, 2...

} On a finite measure space, both notions are equivalent. Otherwise, convergence in measure can refer to either global convergence in measure: 2.2.3 ...

{\displaystyle \mathbb {R} ^{n}} .) A random measure ζ {\displaystyle \zeta } is a (a.s.) locally finite transition kernel from an abstract probability...

of the term is closely related to tightness of a family of measures, since a finite measure μ is inner regular if and only if, for all ε > 0, there is...

measure (also known as a strictly localizable measure) is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite...

denote the Dirac measure centred on some fixed point x in some measurable space (X, Σ). δx is a probability measure, and hence a finite measure. Suppose that...

that μ {\displaystyle \mu } is locally finite, meaning that every point has an open neighborhood with finite measure. For Hausdorff spaces, this implies...

Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a...

{M}},\mu )} is a measure space. Let K ⊂ M {\displaystyle {\mathcal {K}}\subset {\mathfrak {M}}} be a collection of sets of finite measure. A family Φ ⊂ L...

non-negative measure on the Borel σ-algebra of I satisfying μ([a, t]) < ∞ for all t ∈ I (this is certainly satisfied when μ is a locally finite measure). Assume...

finite graph Locally finite group Locally finite measure Locally finite operator in linear algebra Locally finite poset Locally finite space, a topological...

Suppose X and Y are σ-finite measure spaces and suppose that X × Y is given the product measure (which is unique as X and Y are σ-finite). Fubini's theorem...

non-negative measure. To that end, it is a quick check that the real and imaginary parts μ1 and μ2 of a complex measure μ are finite-valued signed measures. One...

satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. Given a field of sets ( Ω...

that any Markov measure on the smaller subshift has a preimage measure that is not Markov of any order (Example 2.6 ). Let V be a finite set of n symbols...

absolute moment is finite, then the r th absolute moment is finite, too. (This also follows from Jensen's inequality.) For two σ-finite measure spaces (S1, Σ1...

subset if the subset has finitely many elements, and infinity ∞ {\displaystyle \infty } if the subset is infinite. The counting measure can be defined on any...

set has zero measure. Since μ(X) = 0, μ is always a finite measure, and hence a locally finite measure. If X is a Hausdorff topological space with its Borel...