Measure Theory & Measurability

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Abstractions about measures, measurable spaces and functions, additivity, completion, products, and non-measurable sets. They include Borel and vector measures, lifting and decomposition, random sets, prevalence, exhaustiveness, and universal measurability.

23 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.

• Atom (measure theory) — Identify a measurable set of positive measure whose measurable subsets have either zero measure or the atom’s full measure.
• Borel measure — A measure defined on the Borel sigma-algebra generated by the open subsets of a topological space.
• Collectively exhaustive events — Require a declared family of events to cover the entire sample space, so every possible outcome belongs to at least one member without requiring the members to be disjoint.
• Complete measure — A measure space in which every subset of every measurable null set is itself measurable and has measure zero.
• Decomposable measure — A measure space partitionable into measurable pieces of finite measure so every measurable set is assembled compatibly from its intersections with those pieces.
• Doob–Dynkin lemma — Characterize when a measurable quantity carries no information beyond a measurable map: under the stated measurable-space conditions it factors as a measurable function of that map exactly when it is measurable with respect to the map's generated sigma-algebra.
• Equivalence (measure theory) — Treat two measures on one measurable space as equivalent exactly when each is absolutely continuous with respect to the other, so they have the same null sets.
• Idempotent measure — A probability measure on a topological group that is unchanged by convolution with itself.
• Lifting theory — The study of selectors that choose pointwise measurable representatives of equivalence classes modulo null sets while preserving algebraic and order structure.
• Loeb space — A standard countably additive measure space constructed from an internal finitely additive measure in nonstandard analysis by taking standard parts and completing the induced measure.
• Measurable space — A set equipped with a sigma-algebra specifying which subsets are admissible as measurable events or regions.
• Non-measurable set — A subset lying outside a specified sigma-algebra, so the chosen measure cannot consistently assign it a value while preserving the measure axioms.
• Pre-measure — A countably additive nonnegative set function defined on an algebra or ring of sets, serving as the extendable precursor of a measure on a generated sigma-algebra.
• Prevalent and shy sets — Translation-based analogues of full measure and measure zero for subsets of infinite-dimensional vector spaces.
• Product measure — Construct a measure on a product measurable space whose values on measurable rectangles multiply the component measures, with existence and uniqueness controlled by sigma-finiteness or related hypotheses.
• Random compact set — A measurable random variable whose values are compact subsets of a complete separable metric space equipped with the Hausdorff topology.
• Standard Borel space — A measurable space isomorphic to the Borel measurable space of a Polish space, providing a regular setting in which measurable bijections and probability constructions behave well.
• Tau additivity — A topological regularity property requiring a measure of any measurable upward-directed union of open sets to equal the supremum of their individual measures.
• Trivial measure — The zero measure on a measurable space, assigning measure zero to every measurable set and serving as the least element under pointwise measure comparison.
• Universally measurable set — A subset of a Polish space measurable in the completion of every finite Borel measure on that space.
• Vector measure — A finitely or countably additive set function taking values in a vector space, typically a Banach space.
• Vitali set — A choice-dependent subset containing one representative from each rational-translation equivalence class in an interval, yielding a canonical example of a non-Lebesgue-measurable set.
• Weakly measurable function — Call a Banach-space-valued function weakly measurable when every scalarization by a continuous linear functional is measurable, and use essential separable-valuedness to determine when this implies strong measurability.