Carathéodory measure

2020 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

The term might refer to different objects in classical measure theory.

Caratheodory measures and outer measures in metric spaces

Consider an outer measure $\mu$ defined on the class $\mathcal{P} (X)$ of subsets of a metric space $(X,d)$. $\mu$ is a Caratheodory outer measure, more often called metric outer measure (cp. with Section 11 of [Ha]), if \begin{equation}\label{e:additive} \mu (A\cup B) = \mu (A) + \mu (B) \end{equation} for every pair of sets $A, B\subset X$ which have positive distance (i.e. such that $\inf \{d(x,y): x\in A, y\in B\} > 0$). A theorem due to Caratheodory shows then that the Borel sets are $\mu$-measurable (see Outer measure#Caratheodory criterion, also for the notion of $\mu$-measurability). The restriction of $\mu$ to the $\sigma$-algebra of $\mu$-measurable sets is called, by some authors, the Caratheodory measure induced by the metric outer measure $\mu$.

The converse is also true: if $\mu$ is an outer measure on a metric space $(X,d)$ for which the open set are $\mu$-measurable, then $\mu$ is a metric outer measure (see for instance Remark (8c) of Section 11 in [Ha]).

Caratheodory outer measures with respect to a class of functions

More generally, given a set $X$ and a class $\Gamma$ of real functions on $X$, some authors (see for instance Section 7 of Chapter 12 in [Ro]) call Caratheodory outer measures with respect to $\Gamma$ those outer measures $\mu$ on $\mathcal{P} (X)$ with the property that \eqref{e:additive} holds when $A$ and $B$ are separated by $\Gamma$, i.e. when there is a function $\varphi\in \Gamma$ with $\inf_A\; \varphi > \sup_B \varphi$ or $\inf_B\;\varphi > \sup_B\; \varphi$.

If $(X,d)$ is a metric space and we chose as $\Gamma$ the set of functions of type $x\mapsto {\rm dist}\, (x, E)$ with $E\subset X$, then a Caratheodory outer measure with respect to $\Gamma$ corresponds to a Caratheodory outer measure in the sense of the previous section.

Caratheodory (outer) measures in the Euclidean space

Some authors use the term Caratheodory (outer) measures for a special class of outer measures defined on the subsets of the euclidean space $\mathbb R^n$ and constructed in a fashion similar to the usual Hausdorff (outer) measures. Cp. for instance with Sections 2.1.3-2.1.4-2.1.5 of [KP] and Sections 2.10.2-2.10.3-2.10.4 of [Fe].

References

[Ca]	C. Carathéodory, "Über das lineare Mass von Punktmengen, eine Verallgemeinerung des Längenbegriffs" Nachr. Gesell. Wiss. Göttingen (1914) pp. 404–426.
[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[KP]	S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
[Ma]	P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Mu]	M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
[Ro]	H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501