• Source: Metric outer measure

In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that




μ
(
A
∪
B
)
=
μ
(
A
)
+
μ
(
B
)


{\displaystyle \mu (A\cup B)=\mu (A)+\mu (B)}


for every pair of positively separated subsets A and B of X.




Construction of metric outer measures


Let τ : Σ → [0, +∞] be a set function defined on a class Σ of subsets of X containing the empty set ∅, such that τ(∅) = 0. One can show that the set function μ defined by




μ
(
E
)
=

lim

δ
→
0



μ

δ


(
E
)
,


{\displaystyle \mu (E)=\lim _{\delta \to 0}\mu _{\delta }(E),}


where





μ

δ


(
E
)
=
inf

{





∑

i
=
1


∞


τ
(

C

i


)

|


C

i


∈
Σ
,
diam
⁡
(

C

i


)
≤
δ
,

⋃

i
=
1


∞



C

i


⊇
E

}

,


{\displaystyle \mu _{\delta }(E)=\inf \left\{\left.\sum _{i=1}^{\infty }\tau (C_{i})\right|C_{i}\in \Sigma ,\operatorname {diam} (C_{i})\leq \delta ,\bigcup _{i=1}^{\infty }C_{i}\supseteq E\right\},}


is not only an outer measure, but in fact a metric outer measure as well. (Some authors prefer to take a supremum over δ > 0 rather than a limit as δ → 0; the two give the same result, since μδ(E) increases as δ decreases.)
For the function τ one can use




τ
(
C
)
=
diam
⁡
(
C

)

s


,



{\displaystyle \tau (C)=\operatorname {diam} (C)^{s},\,}


where s is a positive constant; this τ is defined on the power set of all subsets of X. By Carathéodory's extension theorem, the outer measure can be promoted to a full measure; the associated measure μ is the s-dimensional Hausdorff measure. More generally, one could use any so-called dimension function.
This construction is very important in fractal geometry, since this is how the Hausdorff measure is obtained. The packing measure is superficially similar, but is obtained in a different manner, by packing balls inside a set, rather than covering the set.




Properties of metric outer measures


Let μ be a metric outer measure on a metric space (X, d).

For any sequence of subsets An, n ∈ N, of X with





A

1


⊆

A

2


⊆
⋯
⊆
A
=

⋃

n
=
1


∞



A

n


,


{\displaystyle A_{1}\subseteq A_{2}\subseteq \dots \subseteq A=\bigcup _{n=1}^{\infty }A_{n},}


and such that An and A \ An+1 are positively separated, it follows that




μ
(
A
)
=

sup

n
∈

N



μ
(

A

n


)
.


{\displaystyle \mu (A)=\sup _{n\in \mathbb {N} }\mu (A_{n}).}


All the d-closed subsets E of X are μ-measurable in the sense that they satisfy the following version of Carathéodory's criterion: for all sets A and B with A ⊆ E and B ⊆ X \ E,




μ
(
A
∪
B
)
=
μ
(
A
)
+
μ
(
B
)
.


{\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}


Consequently, all the Borel subsets of X — those obtainable as countable unions, intersections and set-theoretic differences of open/closed sets — are μ-measurable.




References


Rogers, C. A. (1998). Hausdorff measures. Cambridge Mathematical Library (Third ed.). Cambridge: Cambridge University Press. pp. xxx+195. ISBN 0-521-62491-6.

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