View source for Additive function

''finitely-additive function (on sets, on domains)''

A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106501.png" /> defined on a system of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106502.png" /> and such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

for any finite number of pairwise-disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106504.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106505.png" /> whose union also belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106506.png" />. Countably-additive set functions are an important kind of additive functions (cf. [[Countably-additive set function|Countably-additive set function]]).



====Comments====
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106507.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106508.png" />-algebra on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a0106509.png" />. Then a non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a01065010.png" /> (taking, possibly, the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a01065011.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a01065012.png" /> is an additive (finitely-additive, countably-additive) measure if it satisfies (*) for an arbitrary (respectively, finite, countable) number of disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a01065013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010650/a01065014.png" />.

Usually, a measure (sic) is a countably-additive measure.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.L. Royden,   "Real analysis" , Macmillan  (1968)</TD></TR></table>