Measuresuch

Measuresuch is a theoretical construct used in measure theory and probability to denote the process or operator of selecting a measure that satisfies a specified condition, or constraint, expressed as a “such that” statement. The term combines the mathematical notion of a measure with the common phrasing used to impose properties on objects, such as μ such that P(μ).

Formal definition: Let (X, F) be a measurable space and M(X) the set of all probability measures on (X, F). Given a property P that can be stated as a condition on μ ∈ M(X), the measuresuch operation aims to produce a μ ∈ M(X) that satisfies P, provided the set M(P) = { μ ∈ M(X) : μ satisfies P } is nonempty. When several measures fulfill P, the construction may rely on an additional criterion to select a canonical μ.

Properties and variants: Existence and uniqueness depend on the exact constraint P. Measuresuch is typically non-unique without a selection rule. Common choices for selection rules include maximizing entropy, minimizing divergence to a prior measure, or restricting to measures with minimal support. The concept is closely related to constrained optimization and to the study of representative measures under moment or structural constraints.

Examples and applications: In statistics, measuresuch can formalize the idea of choosing a distribution that matches given moments. In Bayesian inference, it resembles selecting priors subject to certain drift or symmetry constraints. In ergodic theory and optimal transport, it appears in the construction of invariant or optimal measures under imposed conditions.

Limitations: The approach can be ill-defined if no measure satisfies P or if the selection rule is inappropriate or computationally intractable. It remains a conceptual tool rather than a standard object in published literature.

See also: measure, probability measure, constrained optimization, maximum entropy, moment problem, selection of measures.