Counting measure

In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ${\displaystyle \infty }$ if the subset is infinite.^[1]

The counting measure can be defined on any measurable space (that is, any set ${\displaystyle X}$ along with a sigma-algebra) but is mostly used on countable sets.^[1]

In formal notation, we can turn any set ${\displaystyle X}$ into a measurable space by taking the power set of ${\displaystyle X}$ as the sigma-algebra ${\displaystyle \Sigma ;}$ that is, all subsets of ${\displaystyle X}$ are measurable sets. Then the counting measure ${\displaystyle \mu }$ on this measurable space ${\displaystyle (X,\Sigma )}$ is the positive measure ${\displaystyle \Sigma \to [0,+\infty ]}$ defined by

for all ${\displaystyle A\in \Sigma ,}$ where ${\displaystyle \vert A\vert }$ denotes the cardinality of the set ${\displaystyle A.}$^[2]

The counting measure on ${\displaystyle (X,\Sigma )}$ is σ-finite if and only if the space ${\displaystyle X}$ is countable.^[3]

Integration on ${\displaystyle \mathbb {N} }$ with counting measure

Take the measure space ${\displaystyle (\mathbb {N} ,2^{\mathbb {N} },\mu )}$, where ${\displaystyle 2^{\mathbb {N} }}$ is the set of all subsets of the naturals and ${\displaystyle \mu }$ the counting measure. Take any measurable ${\displaystyle f:\mathbb {N} \to [0,\infty ]}$. As it is defined on ${\displaystyle \mathbb {N} }$, ${\displaystyle f}$ can be represented pointwise as

Each ${\displaystyle \phi _{M}}$ is measurable. Moreover ${\displaystyle \phi _{M+1}(x)=\phi _{M}(x)+f(M+1)\cdot 1_{\{M+1\}}(x)\geq \phi _{M}(x)}$. Still further, as each ${\displaystyle \phi _{M}}$ is a simple function

Hence by the monotone convergence theorem

Discussion

The counting measure is a special case of a more general construction. With the notation as above, any function ${\displaystyle f:X\to [0,\infty )}$ defines a measure ${\displaystyle \mu }$ on ${\displaystyle (X,\Sigma )}$ via

where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is, Taking ${\displaystyle f(x)=1}$ for all ${\displaystyle x\in X}$ gives the counting measure.

See also

• Pip (counting) – Easily countable items
• Set function – Function from sets to numbers

References