Måttkvantifierare

Måttkvantifierare is a term used in Swedish-language discussions of logic and measure theory to refer to quantifiers that range over measures on a fixed measurable space. In contrast to ordinary quantified variables that range over elements of a domain, a måttkvantifierare binds a variable whose interpretation is a measure—for example a probability measure or a sigma-additive measure on a given σ-algebra.

Formalization is not standardized. A typical approach introduces a separate sort M for measures on (X, Σ). The language is extended with quantifiers ∀μ ∈ M and ∃μ ∈ M and allows formulas that mention μ and measurable sets A ⊆ X. Semantics are given relative to a structure that includes the measure space (X, Σ) together with a chosen class of measures. Example: ∃μ ∈ M [ μ is a probability measure ∧ μ(A) > 0 ] asserts the existence of a measure with positive mass on A. Another example: ∀μ ∈ M [ μ(X) = 1 → φ(μ) ], when μ is restricted to probability measures.

Relation and use: Measure quantifiers appear in probabilistic and statistical logics, descriptive set theory, and certain branches of model theory where one studies properties that hold for families of measures rather than for individual elements. They connect with randomized or probabilistic semantics and with continuous logic's emphasis on metric structures. They are largely a topic of theoretical investigation and are not universally standardized; different authors may define the syntax and semantics differently or use alternative terminology such as "quantifier over measures" or "measure quantifiers."

See also: Measure theory; Quantifier (logic); Probabilistic logic; Continuous logic; Model theory.