lessthanadditive

Lessthanadditive is a term used in measure theory to describe a class of set functions that satisfy a relaxed additivity condition. In its basic form, a function f defined on the power set of a set X is called lessthanadditive if for any disjoint sets A and B, the inequality f(A ∪ B) ≤ f(A) + f(B) holds. This captures the idea of subadditivity restricted to unions of disjoint sets, though the term is sometimes used informally to emphasize the inequality rather than equality.

Relationship to other concepts: A function that is additive satisfies f(A ∪ B) = f(A) + f(B) for all disjoint A and B, and such a function is automatically lessthanadditive. Conversely, a lessthanadditive function need not be additive. The concept aligns closely with subadditivity, which typically states f(A ∪ B) ≤ f(A) + f(B) for any A and B (not necessarily disjoint); many texts use subadditivity as the foundational property, with lessthanadditive serving as a descriptive label in specific contexts.

Examples: The Lebesgue outer measure μ* is subadditive, and therefore lessthanadditive, since μ*(A ∪ B) ≤ μ*(A) + μ*(B) for any A and B. Capacities and other non-additive set functions that are monotone and satisfy f(A ∪ B) ≤ f(A) + f(B) for disjoint A and B are also described as lessthanadditive in some sources. In practice, the term helps distinguish functions that bound unions from the sum of parts without requiring exact additivity.

Applications and notes: Lessthanadditive properties are useful for proving upper bounds on measures of unions, in risk assessment models, and in theories where exact additivity is too strong a requirement. The term is not universally standardized and is often used interchangeably with subadditivity in literature, depending on the author.

See also: additive set function, subadditive, outer measure, capacity.