Hölder's inequality

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.

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

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

$\mu (A)=\{\begin{array}{ll}\left|A\right|& \text{if }A\text{ is finite}\\ +\mathrm{\infty }& \text{if }A\text{ is infinite}\end{array}$

for all $A\in \mathrm{\Sigma }$, where $\left|A\right|$ denotes the cardinality of the set $A$.

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

Notes

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References

• 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.