Counting measure

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In mathematics, 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 if the subset is infinite.[1]

The counting measure can be defined on any measurable set, but is mostly used on countable sets.[1]

In formal notation, we can make any set X into a measurable space by taking the sigma-algebra \Sigma of measurable subsets to consist of all subsets of X. Then the counting measure \mu on this measurable space (X,\Sigma) is the positive measure \Sigma\rightarrow[0,+\infty] defined by


\mu(A)=\begin{cases}
\vert A \vert & \text{if } A \text{ is finite}\\
+\infty & \text{if } A \text{ is infinite}
\end{cases}

for all A\in\Sigma, where \vert A\vert denotes the cardinality of the set A.[2]

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

Discussion[edit]

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

\mu(A \subseteq X):=\sum_{a \in A} f(a),

where the possibly uncountable sum of real numbers is defined to be the sup of the sums over all finite subsets, i.e.,

\sum_{y \in Y \subseteq \mathbb R} y := \sup_{F \subseteq Y, |F| < \infty} \left\{ \sum_{y \in F} y  \right\}.

Taking f(x)=1 for all x in X produces the counting measure.

Notes[edit]

  1. ^ a b Counting Measure at PlanetMath.org.
  2. ^ Schilling (2005), p.27
  3. ^ Hansen (2009) p.47

References[edit]

  • Schilling, René L. (2005)."Measures, Integral and Martingales". Cambridge University Press.
  • Hansen, Ernst (2009)."Measure theory, Fourth Edition". Department of Mathematical Science, University of Copenhagen.