Measure (source code)
= Measure
{title2=$\mu$}
{wiki}
On a <measurable space> $(X,\mathcal F)$, a measure is a <function> $\mu:\mathcal F\to[0,\infty]$ with $\mu(\varnothing)=0$ and <countable additivity>: for pairwise disjoint <measurable sets> $A_n$,
$$
\mu\left(\bigcup_{n=1}^{\infty}A_n\right)=\sum_{n=1}^{\infty}\mu(A_n).
$$
Infinite size is permitted. Nonnegativity implies that if <measurable sets> satisfy $A\subseteq B$, then $\mu(A)\leq\mu(B)$: decompose $B$ into $A$ and $B\setminus A$. For overlapping <measurable sets>, use <countable subadditivity> rather than adding their full sizes twice.
For example, <counting measure> gives $\mu(\{a,b,c\})=3$, while a <Dirac measure> $\delta_a$ gives that same <set> size $1$ if it contains $a$. Under <Lebesgue measure>, $[0,1/4]\cup[3/4,1]$ has size $1/4+1/4=1/2$. These are all <measures>, but they answer different questions: how many objects, whether a distinguished object is present, or how much length is occupied.
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