Editing Sigma-additive set function (section)

==Properties==

Useful properties of an additive set function <math>\mu</math> include the following.

===Value of empty set===

Either <math>\mu(\varnothing) = 0,</math> or <math>\mu</math> assigns <math>\infty</math> to all sets in its domain, or <math>\mu</math> assigns <math>- \infty</math> to all sets in its domain. ''Proof'': additivity implies that for every set <math>A,</math> <math>\mu(A) = \mu(A \cup \varnothing) = \mu(A) + \mu( \varnothing)</math> (it's possible in the edge case of an empty domain that the only choice for <math>A</math> is the [[empty set]] itself, but that still works). If <math>\mu(\varnothing) \neq 0,</math> then this equality can be satisfied only by plus or minus infinity.

===Monotonicity===

If <math>\mu</math> is non-negative and <math>A \subseteq B</math> then <math>\mu(A) \leq \mu(B).</math> That is, <math>\mu</math> is a '''{{visible anchor|monotone set function}}'''. Similarly, If <math>\mu</math> is non-positive and <math>A \subseteq B</math> then <math>\mu(A) \geq \mu(B).</math> 

===Modularity{{Anchor|modularity}}===
{{See also|Valuation (geometry)}}
{{See also|Valuation (measure theory)}}

A [[set function]] <math>\mu</math> on a [[family of sets]] <math>\mathcal{S}</math> is called a '''{{visible anchor|modular set function}}''' and a '''[[Valuation (geometry)|{{visible anchor|valuation}}]]''' if whenever <math>A,</math> <math>B,</math> <math>A\cup B,</math> and <math>A\cap B</math> are elements of <math>\mathcal{S},</math> then
<math display=block> \phi(A\cup B)+  \phi(A\cap B) = \phi(A) + \phi(B)</math> 
The above property is called '''{{visible anchor|modularity}}''' and the argument below proves that additivity implies modularity.

Given <math>A</math> and <math>B,</math> <math>\mu(A \cup B) + \mu(A \cap B) = \mu(A) + \mu(B).</math> ''Proof'': write <math>A = (A \cap B) \cup (A \setminus B)</math> and <math>B = (A \cap B) \cup (B \setminus A)</math> and <math>A \cup B = (A \cap B) \cup (A \setminus B) \cup (B \setminus A),</math> where all sets in the union are disjoint. Additivity implies that both sides of the equality equal <math>\mu(A \setminus B) + \mu(B \setminus A) + 2\mu(A \cap B).</math>

However, the related properties of [[Submodular set function|''submodularity'']] and [[Subadditive set function|''subadditivity'']] are not equivalent to each other.

Note that modularity has a different and unrelated meaning in the context of complex functions; see [[modular form]].

===Set difference===

If <math>A \subseteq B</math> and <math>\mu(B) - \mu(A)</math> is defined, then <math>\mu(B \setminus A) = \mu(B) - \mu(A).</math>