Invariant Measures & Ergodic Probability

← Back to Domain-Specific Families

Abstractions about invariant, discrete, random, pushforward, and sub-probability measures, along with ergodicity, recurrence, moments, and equidistribution.

12 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.

• Bertrand paradox (probability) — A geometric-probability paradox in which different seemingly natural random-chord constructions produce different answers, revealing that randomness requires a specified measure.
• 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.
• Equidistributed sequence — Require the limiting frequency of sequence terms in every subinterval to equal that subinterval’s normalized length.
• Ergodicity — A measure-preserving dynamical property in which every invariant measurable set has measure zero or full measure, making the system statistically indecomposable.
• Factorial moment — Take the expectation of a random variable’s falling factorial to expose counting structure and probability-generating-function derivatives.
• 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.
• Poisson boundary — A measure-theoretic boundary of a random walk that captures its asymptotic tail behavior and represents bounded harmonic functions by boundary data.
• 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.
• 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.
• Recurrent point — A point of a dynamical system that returns arbitrarily close to itself at arbitrarily late iterates, equivalently belonging to its own omega-limit set.
• Reynolds operator — Project objects onto their invariant part by averaging over a symmetry group or averaging regime, with linearity, idempotence, and compatibility rules governing the result.
• 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.