partialmeasure

Partialmeasure refers to a set function that behaves like a measure but is defined only on a subset of the power set of a given set. In practice, the domain is a collection of sets such as a ring or an algebra (or, less commonly, another suitable family closed under finite unions and, in some cases, complements). The function μ takes values in [0, ∞] and satisfies the basic measure-like properties on its domain: monotonicity (A ⊆ B implies μ(A) ≤ μ(B)) and additivity on disjoint families whose union remains in the domain.

When the domain is an algebra and μ is additive for finite disjoint unions and countably additive for disjoint sequences contained in the domain, μ is often called a premeasure. A partial measure defined on a ring or algebra may be finite on its domain, which facilitates extension.

Carathéodory’s extension theorem provides a standard pathway from partial measures (premeasures) to full measures. If μ is a nonnegative, countably additive set function defined on a ring or algebra and appropriate finiteness or σ-finiteness conditions hold, μ can be extended uniquely to a measure on the σ-algebra generated by the domain. The resulting measure agrees with μ on the original sets.

Partial measures arise in various contexts, such as constructing measures from simple blocks (intervals, rectangles, or finite unions thereof), or in probability theory where probabilities are initially assigned to a subcollection of events. They also connect to outer measures and the Carathéodory construction, which often broadens a partial measure to a complete measure on a larger σ-algebra.

In summary, a partialmeasure is a measure-like function defined on a limited domain, with the potential to extend to a full measure under suitable extension theorems.