Counting measure

 ( Measures ) 

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 symbol if the subset is Infinite set.

The counting measure can be defined on any measurable space (that is, any set $X$ along with a sigma-algebra) but is mostly used on countable sets.

In formal notation, we can turn any set $X$ into a measurable space by taking the power set of $X$ as the sigma-algebra $\backslash Sigma;$ that is, all subsets of $X$ are measurable sets. Then the counting measure $\backslash mu$ on this measurable space $(X,\backslash Sigma)$ is the positive measure $\backslash Sigma\; \backslash to\; 0,+\backslash infty$ defined by for all $A\backslash in\backslash Sigma,$ where $\backslash vert\; A\backslash vert$ denotes the cardinality of the set $A.$

The counting measure on $(X,\backslash Sigma)$ is σ-finite if and only if the space $X$ is countable.

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Integration on the set of natural numbers with counting measure Take the measure space $(\backslash mathbb\{N\},\; 2^\backslash mathbb\{N\},\; \backslash mu)$, where $2^\backslash mathbb\{N\}$ is the set of all subsets of the naturals and $\backslash mu$ the counting measure. Take any measurable $f\; :\; \backslash mathbb\{N\}\; \backslash to\; 0,\backslash infty$. As it is defined on $\backslash mathbb\{N\}$, $f$ can be represented pointwise as $$f(x)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; f(n)\; 1\_\{\backslash \{n\backslash \}\}(x)\; =\; \backslash lim\_\{M\; \backslash to\; \backslash infty\}\; \backslash underbrace\{\; \backslash \; \backslash sum\_\{n=1\}^M\; f(n)\; 1\_\{\backslash \{n\backslash \}\}(x)\; \backslash \; \}\_\{\; \backslash phi\_M\; (x)\; \}\; =\; \backslash lim\_\{M\; \backslash to\; \backslash infty\}\; \backslash phi\_M\; (x)$$

Each $\backslash phi\_M$ is measurable. Moreover $\backslash phi\_\{M+1\}(x)\; =\; \backslash phi\_M\; (x)\; +\; f(M+1)\; \backslash cdot\; 1\_\{\; \backslash \{M+1\backslash \}\; \}(x)\; \backslash geq\; \backslash phi\_M\; (x)$. Still further, as each $\backslash phi\_M$ is a simple function $$\backslash int\_\backslash mathbb\{N\}\; \backslash phi\_M\; d\backslash mu\; =\; \backslash int\_\backslash mathbb\{N\}\; \backslash left(\; \backslash sum\_\{n=1\}^M\; f(n)\; 1\_\{\backslash \{n\backslash \}\}\; (x)\; \backslash right)\; d\backslash mu\; =\; \backslash sum\_\{n=1\}^M\; f(n)\; \backslash mu\; (\backslash \{n\backslash \})\; =\; \backslash sum\_\{n=1\}^M\; f(n)\; \backslash cdot\; 1\; =\; \backslash sum\_\{n=1\}^M\; f(n)$$Hence by the monotone convergence theorem $$\backslash int\_\backslash mathbb\{N\}\; f\; d\backslash mu\; =\; \backslash lim\_\{M\; \backslash to\; \backslash infty\}\; \backslash int\_\backslash mathbb\{N\}\; \backslash phi\_M\; d\backslash mu\; =\; \backslash lim\_\{M\; \backslash to\; \backslash infty\}\; \backslash sum\_\{n=1\}^M\; f(n)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; f(n)$$

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Discussion The counting measure is a special case of a more general construction. With the notation as above, any function $f\; :\; X\; \backslash to\; [0,\; \backslash infty)$ defines a measure $\backslash mu$ on $(X,\; \backslash Sigma)$ via $$\backslash mu(A):=\backslash sum\_\{a\; \backslash in\; A\}\; f(a)\backslash quad\; \backslash text\{\; for\; all\; \}\; A\; \backslash subseteq\; X,$$ where the possibly uncountable sum of real numbers is defined to be the supremum of the sums over all finite subsets, that is, Taking $f(x)\; =\; 1$ for all $x\; \backslash in\; X$ gives the counting measure.

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See also

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