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In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.[1][2] Carathéodory's work on outer measures found many applications in measure-theoretic set theory (outer measures are for example used in the proof of the fundamental Carathéodory's extension theorem), and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.
Measures are generalizations of length, area and volume, but are useful for much more abstract and irregular sets than intervals in or balls in . One might expect to define a generalized measuring function on that fulfills the following requirements:
Any interval of reals has measure
The measuring function is a non-negative extended real-valued function defined for all subsets of .
Translation invariance: For any set and any real , the sets and have the same measure
Countable additivity: for any sequence of pairwise disjoint subsets of
It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.
In the mathematical field of measure theory, an outermeasure or exterior measure is a function defined on all subsets of a given set with values in the...
subset E ⊆ R {\displaystyle E\subseteq \mathbb {R} } , the Lebesgue outermeasure λ ∗ ( E ) {\displaystyle \lambda ^{\!*\!}(E)} is defined as an infimum...
In mathematics, a metric outermeasure is an outermeasure μ defined on the subsets of a given metric space (X, d) such that μ ( A ∪ B ) = μ ( A ) + μ...
on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with...
In mathematics, an outermeasure μ on n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold: Every Borel...
specifically fractals and their Hausdorff dimensions. It is a type of outermeasure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set...
via the Lebesgue measure. In the one-dimensional case, the Lebesgue outermeasure of a set is defined in terms of the lengths of open intervals. Concretely...
points have measure 0 {\displaystyle 0} , so the measure of any compact subset of this vertical segment is 0 {\displaystyle 0} . But, using outer regularity...
inner regular. A measure is called outer regular if every measurable set is outer regular. A measure is called regular if it is outer regular and inner...
outer regular. a Borel regular measure if it is a Borel measure that is also regular. a Radon measure if it is a regular and locally finite measure....
complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero)...
first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outermeasure is constructed: Let X {\displaystyle...
Equivalently, a cardinal number α is an Ulam number if whenever ν is an outermeasure on a set Y, and F a set of pairwise disjoint subsets of Y, ν(⋃F) < ∞...
{\displaystyle C} . If a Borel measure μ {\displaystyle \mu } is both inner regular and outer regular, it is called a regular Borel measure. If μ {\displaystyle...
Xn of a common set X, where X has a Carathéodory outermeasure and each Xi has finite outermeasure. Their first general formulation is as follows: for...
measures are often used in combination with outermeasures to extend a measure to a larger σ-algebra. If μ {\displaystyle \mu } is a finite measure defined...
Outer space (or simply space) is the expanse beyond celestial bodies and their atmospheres. It contains ultra-low levels of particle densities, constituting...
A tape measure or measuring tape is a flexible ruler used to measure length or distance. It consists of a ribbon of cloth, plastic, fibre glass, or metal...
Dimension and outermeasure from 1919 is particularly outstanding. In this work, the concepts were introduced which are now known as Hausdorff measure and the...
semantics of evaluation Valuation (measure theory), a tool for constructing outermeasures Valuation (ethics), the determination of the ethic or philosophic value...
Non-measurable set – Set which cannot be assigned a meaningful "volume" Outermeasure – Mathematical function Infinite parity function – Boolean function...
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