Measure & Probability Foundations

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Abstractions about the foundational objects of measure theory and probability — measurable spaces and sigma-algebras, types of measures (Borel, invariant, product, vector), measure-theoretic pathologies (non-measurable sets, Vitali sets), and core probability results such as the probability axioms and the Radon-Nikodym theorem.

42 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.
• Caccioppoli set — A measurable set whose characteristic function has locally bounded variation, equivalently a set of locally finite perimeter in the geometric-measure-theory sense.
• 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.
• Complementary event — For an event in a sample space, the event containing exactly the outcomes in the sample space that are not in the original event.
• 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.
• Discrete measure — A measure concentrated on an at most countable set, representable as a countable weighted sum of point masses under the stated measurable-space convention.
• 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.
• Ergodicity — A measure-preserving dynamical property in which every invariant measurable set has measure zero or full measure, making the system statistically indecomposable.
• Idempotent measure — A probability measure on a topological group that is unchanged by convolution with itself.
• Imprecise probability — Represent incomplete probabilistic commitment by a coherent set of admissible probability measures or equivalent lower and upper expectations, so conclusions expose a range and distinguish robust agreement from decisions that depend on an unresolved model choice.
• Invariant measure — A measure preserved by a specified transformation or group action, assigning every measurable set the same measure as its preimage or transformed image.
• Law of total probability — A probability identity expressing an event's probability as the sum or integral of its conditional probabilities over a mutually exclusive exhaustive partition.
• 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.
• Lévy–Prokhorov metric — A distance between probability measures on a metric space that permits both spatial enlargement and a matching probability slack, metrizing weak convergence on separable spaces and connecting compactness to tightness.
• Measurable space — A set equipped with a sigma-algebra specifying which subsets are admissible as measurable events or regions.
• Metric outer measure — An outer measure additive on sets separated by a positive distance in a metric space.
• 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.
• Poisson boundary — A measure-theoretic boundary of a random walk that captures its asymptotic tail behavior and represents bounded harmonic functions by boundary data.
• 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.
• Probability axioms — The foundational conditions requiring a probability measure to be nonnegative, assign one to the whole sample space and add over countably many disjoint events.
• Probability measure — A countably additive measure on a sigma-algebra that assigns total mass one to the sample space.
• 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.
• Radon–Nikodym set — The convex range of a vector measure formed from several agents' nonatomic valuations of measurable cake pieces, representing all simultaneously attainable value vectors.
• Radon–Nikodym theorem — A measure-theoretic theorem representing a sigma-finite measure absolutely continuous with respect to another as integration against an almost-everywhere unique density.
• Random compact set — A measurable random variable whose values are compact subsets of a complete separable metric space equipped with the Hausdorff topology.
• Random measure — A measure-valued random element or kernel that assigns each outcome a locally finite measure, unifying random point configurations and stochastic mass distributions.
• 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.
• Sub-probability measure — A nonnegative countably additive measure whose total mass is at most one, allowing missing mass to represent termination, failure or an unmodeled outcome.
• Tangent measure — A weak limit of rescaled blow-ups of a Radon measure around a point, capturing its infinitesimal mass geometry.
• 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.
• Tightness of measures — The property that probability mass can be captured uniformly well inside compact subsets.
• 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.
• Unit measure — The probability axiom requiring the measure of the entire sample space to equal one.
• 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 covering lemma — A geometric selection lemma extracting pairwise disjoint balls from a family so that a fixed enlargement of the selected balls covers the original union or set of centers.
• 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.