View source for Probability measure

''probability distribution, probability''

{{MSC|60-01}}

A real non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749301.png" /> on a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749302.png" /> of subsets (events) of a non-empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749303.png" /> (the space of elementary events) forming a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749305.png" />-field (i.e. a set closed with respect to countable set-theoretic operations) 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/p/p074/p074930/p0749306.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749307.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749308.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p0749309.png" />-additivity).

===Examples of probability measures.===


1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493010.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493011.png" /> is the class of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493013.png" /> (this probability measure corresponds to a random experiment consisting in throwing a symmetrical coin; if heads correspond to 1 while tails correspond to 2, the probability of throwing heads (tails) is 1/2);

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493014.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493015.png" /> is the class of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493016.png" />;

<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/p/p074/p074930/p07493017.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493018.png" /> (the [[Poisson distribution|Poisson distribution]]);

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493019.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493020.png" /> is the class of Borel subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493021.png" />;

<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/p/p074/p074930/p07493022.png" /></td> </tr></table>

(the [[Normal distribution|normal distribution]]);

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493023.png" /> is the space of continuous real functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493025.png" /> that vanish at the point zero; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493026.png" /> is the class of Borel subsets with respect to the topology of uniform convergence; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493027.png" /> is the measure which is uniquely defined by the formula

<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/p/p074/p074930/p07493028.png" /></td> </tr></table>

<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/p/p074/p074930/p07493029.png" /></td> </tr></table>

<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/p/p074/p074930/p07493030.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493031.png" /> is an arbitrary natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074930/p07493032.png" /> (the [[Wiener measure|Wiener measure]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian) {{MR|0032961}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> B.V. Gnedenko,   [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint (1962) (Translated from Russian) </TD></TR></table>

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Billingsley, "Probability and measure" , Wiley (1979) {{MR|0534323}} {{ZBL|0411.60001}} </TD></TR></table>