Measure-Theoretic Constructions

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Abstractions about generalized measures and integration on abstract spaces, covering measure types (Borel Regular Measure, Counting Measure, Locally Finite Measure, Fuzzy Measure Theory), integration theories (Pettis Integral, Projection-Valued Measure), and geometric-measure-theory notions like Spherical Measure and Uniformly Distributed Measure.

10 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Borel regular measure — In mathematics, an outer measure μ on n-dimensional Euclidean space R n is called a Borel regular measure if the following two conditions hold.
• Counting measure — The counting measure can be defined on any measurable space (that is, any set X along with a sigma-algebra) but is mostly used on countable sets.
• Fuzzy measure theory — In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity.
• Locally finite measure — In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.
• Multifractal system — A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.
• Pettis integral — Integrate a Banach-space-valued function weakly by requiring every continuous linear functional to yield an ordinary scalar integral represented by one vector for each measurable set.
• Projection-valued measure — In mathematics, particularly in functional analysis, a projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space.
• Souček space — In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician Jiří Souček.
• Spherical Measure — In mathematics — specifically, in geometric measure theory — spherical measure σ n is the "natural" Borel measure on the n-sphere S n .
• Uniformly distributed measure — In mathematics — specifically, in geometric measure theory — a uniformly distributed measure on a metric space is one for which the measure of an open ball depends only on its radius and not on its centre.