View source for Field of sets

A collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200601.png" /> of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200602.png" /> satisfying:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200603.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200604.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200605.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200607.png" />.

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f1200609.png" />-field of sets is a field of sets satisfying in addition

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006011.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006013.png" />.

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006014.png" />-field is also sometimes called a [[Borel field of sets|Borel field of sets]].

Sometimes, an algebra (respectively, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006015.png" />-algebra) of sets is taken to mean a field (respectively, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120060/f12006016.png" />-field) of sets.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Loeve,   "Probability theory" , v. Nostrand  (1963)  pp. 59  (Edition: Third)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bauer,   "Probability theory and elements of measure theory" , Holt, Rinehart&amp;Winston  (1972)  pp. 7</TD></TR></table>