Measure (mathematics)

Non-negativity: For all ${\displaystyle E\in \Sigma ,\ \ \mu (E)\geq 0.}$

${\displaystyle \mu (\varnothing )=0.}$

Countable additivity (or ): For all countable collections ${\displaystyle \{E_{k}\}_{k=1}^{\infty }}$ of pairwise disjoint sets in Σ,

$${\displaystyle \mu \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu (E_{k}).}$$

${\displaystyle \sigma }$-additivity

The is defined by ${\displaystyle \mu (S)}$ = number of elements in ${\displaystyle S.}$

counting measure

The on ${\displaystyle \mathbb {R} }$ is a complete translation-invariant measure on a σ-algebra containing the intervals in ${\displaystyle \mathbb {R} }$ such that ${\displaystyle \mu ([0,1])=1}$; and every other measure with these properties extends the Lebesgue measure.

Lebesgue measure

Circular measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.

angle

The for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.

Haar measure

The is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.

Hausdorff measure

Every gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure or distribution. See the list of probability distributions for instances.

probability space

The δ_a (cf. Dirac delta function) is given by δ_a(S) = χ_S(a), where χ_S is the indicator function of ${\displaystyle S.}$ The measure of a set is 1 if it contains the point ${\displaystyle a}$ and 0 otherwise.

Dirac measure

Every sigma-finite measure is semifinite.

[5]

Let ${\displaystyle d}$ be a complete, separable metric on ${\displaystyle X,}$ let ${\displaystyle {\cal {B}}}$ be the Borel sigma-algebra induced by ${\displaystyle d,}$ and let ${\displaystyle s\in \mathbb {R} _{>0}.}$ Then the ${\displaystyle {\cal {H}}^{s}|{\cal {B}}}$ is semifinite.^[6]

Hausdorff measure

Let ${\displaystyle d}$ be a complete, separable metric on ${\displaystyle X,}$ let ${\displaystyle {\cal {B}}}$ be the Borel sigma-algebra induced by ${\displaystyle d,}$ and let ${\displaystyle s\in \mathbb {R} _{>0}.}$ Then the ${\displaystyle {\cal {H}}^{s}|{\cal {B}}}$ is semifinite.^[7]

packing measure

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Measure"

Tutorial: Measure Theory for Dummies