Subadditive function

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 collections of subset 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 $$ f\left({ \cup_{i=1}^\infty A_i }\right) \le \sum_{i=1}^\infty f(A_i) \ . $$