# Subadditive function

From Encyclopedia of Mathematics

A real function $f$ with the property $$ f(x+y) \le f(x) + f(y) \ . $$

A subadditive set function is a function $f$ on a collection of subsets of a set $X$ with the property that
$$
f(A \cup B) \le f(A) + f(B) \ .
$$
A set function is $\sigma$-*subadditive* or *countably subadditive* if
$$
f\left({ \cup_{i=1}^\infty A_i }\right) \le \sum_{i=1}^\infty f(A_i) \ .
$$

**How to Cite This Entry:**

Subadditive function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Subadditive_function&oldid=39139