Measure Theory

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20 domain-specific abstractions whose origin domain is Measure Theory.

• 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.
• 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.
• 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.
• Lifting theory — The study of selectors that choose pointwise measurable representatives of equivalence classes modulo null sets while preserving algebraic and order structure.
• 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.
• 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.
• 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.
• Pushforward measure — The measure on a target measurable space obtained by assigning each target set the original measure of its preimage under a measurable map.
• 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.
• 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.
• Vague topology — A topology on Radon measures defined by convergence of integrals against a declared class of continuous test functions, making local mass behavior observable while allowing mass to escape to infinity.
• 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.