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. 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 R {\displaystyle \mathbb {R} } or balls in R 3 {\displaystyle \mathbb {R} ^{3}} . One might expect to define a generalized measuring function φ {\displaystyle \varphi } on R {\displaystyle \mathbb {R} } that fulfills the following requirements:
Any interval of reals [ a , b ] {\displaystyle [a,b]} has measure b − a {\displaystyle b-a}
The measuring function φ {\displaystyle \varphi } is a non-negative extended real-valued function defined for all subsets of R {\displaystyle \mathbb {R} } . Translation invariance: For any set A {\displaystyle A} and any real x {\displaystyle x} , the sets A {\displaystyle A} and A + x = { a + x : a ∈ A } {\displaystyle A+x=\{a+x:a\in A\}} have the same measure Countable additivity: for any sequence ( A j ) {\displaystyle (A_{j})} of pairwise disjoint subsets of R {\displaystyle \mathbb {R} }
φ ( ⋃ i = 1 ∞ A i ) = ∑ i = 1 ∞ φ ( A i ) . {\displaystyle \varphi \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\varphi (A_{i}).}
It turns out that these requirements are incompatible conditions; see non-measurable set. The purpose of constructing an outer measure on all subsets of X {\displaystyle X} is to pick out a class of subsets (to be called measurable) in such a way as to satisfy the countable additivity property.
Outer measures Given a set X , {\displaystyle X,} let 2 X {\displaystyle 2^{X}} denote the collection of all subsets of X , {\displaystyle X,} including the empty set ∅ . {\displaystyle \varnothing .} An outer measure on X {\displaystyle X} is a set function
μ : 2 X → [ 0 , ∞ ] {\displaystyle \mu :2^{X}\to [0,\infty ]}
such that
null empty set: μ ( ∅ ) = 0 {\displaystyle \mu (\varnothing )=0}
countably subadditive: for arbitrary subsets A , B 1 , B 2 , … {\displaystyle A,B_{1},B_{2},\ldots } of X , {\displaystyle X,}
if A ⊆ ⋃ j = 1 ∞ B j then μ ( A ) ≤ ∑ j = 1 ∞ μ ( B j ) . {\displaystyle {\text{if }}A\subseteq \bigcup _{j=1}^{\infty }B_{j}{\text{ then }}\mu (A)\leq \sum _{j=1}^{\infty }\mu (B_{j}).}
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