subadditive

Subadditive is a term used in mathematics to describe a property of functions, sequences, or set functions. A function f: D → R is subadditive if for all x and y in D with x + y in D, f(x + y) ≤ f(x) + f(y). In the special case of sequences (a_n), subadditivity means a_{m+n} ≤ a_m + a_n for all m, n.

Examples: The absolute value function is subadditive because |x + y| ≤ |x| + |y| for all x and y. Outer measures are subadditive: μ*(A ∪ B) ≤ μ*(A) + μ*(B). The Shannon entropy H satisfies H(X, Y) ≤ H(X) + H(Y) for any random variables X and Y. The concept also appears in cost functions and in risk measures, where a subadditive risk measure ρ satisfies ρ(X + Y) ≤ ρ(X) + ρ(Y).

Key results: For subadditive sequences (a_n), Fekete's lemma states that the limit lim_{n→∞} a_n/n exists and equals inf a_n/n. This provides a well-defined growth rate and is used in the analysis of asymptotic behavior.

See also: Submultiplicativity, convexity, measure theory.

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