= Measure space
{title2=$(X,\mathcal M,\mu)$}
{wiki}
A measure space is a <set> $X$, a <sigma-algebra> $\mathcal M$ of its <subsets>, and a <measure> $\mu$ defined on $\mathcal M$. Specifying only $X$ leaves both the measurable questions and their weights undetermined.
For a concrete <probability space>, take $X=\{a,b,c\}$, $\mathcal M$ its <power set>, and weights $1/2,1/3,1/6$. Define $\mu(A)$ by adding the weights of the elements in $A$. Then
$$
\mu(X)=\frac12+\frac13+\frac16=1,\qquad
\mu(\{b,c\})=\frac12.
$$
A <probability measure> is exactly a <measure> normalized by $\mu(X)=1$, usually written $\mathbb P(\Omega)=1$. An <indicator function> has <expected value> equal to its <event>'s <probability>. For a different continuous example, $[0,1]$ with the restricted <Lebesgue measure> is a <probability space>; the <event> $[0,1/4]$ has <probability> $1/4$.
Back to article page
