Namespaces
Variants
Actions

Difference between revisions of "Subadditive function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Start article: Subadditive function)
 
m (better)
 
(One intermediate revision by the same user not shown)
Line 2: Line 2:
 
$$
 
$$
 
f(x+y) \le f(x) + f(y) \ .
 
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) \ .
 
$$
 
$$

Latest revision as of 06:33, 17 September 2016

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=39134