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Tried to find reference containg proof of the following theorem due to Caratheodory:

  • Let: $(X,\rho)$ - metric space. Then:
  • $\mu^{*}$ is metric $\iff \mathcal B(X) \subset \mathfrak M(\mu^{*})$

I would be very thankful if anyone would mention where the proof of this theorem can be found. Any help very appreciated!

  • 1
    Can you define what you mean by all these terms? B(X) looks like the borel sigma algebra but what are the other terms? – 2017-01-28
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    B(X) - borel sigma algebra – 2017-01-28
  • 0
    \mu^{+} - outer measure – 2017-01-28
  • 0
    M(\mu^{*}) - the family of subsets of \mu^{*} – 2017-01-28
  • 1
    I don't understand M(\mu^{}). Do you mean to say that is the space of all \mu* measurable sets? Also what does it mean by an outer measure being a metric? that does not make any sense – 2017-01-28
  • 1
    This reminds me of the fundamental result $\mathfrak{Z^{\mathfrak{k}}} \iff \mathfrak{JH}$, which holds for suitable definitions of $\mathfrak{Z^{\mathfrak{k}}} $ and $\mathfrak{JH}$. – 2017-01-28

1 Answers 1

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If you mean that if $\mu^*$ is a metric outer measure, then all Borel sets are $\mu^*$-measureable, the answer is yes:

Since $\mu^*$ is a metric outer measure, $\mu^∗(E ∪ F) = \mu^∗(E) + \mu^∗(F)$ whenever dist $(E, F) > 0.$

Let $U$ be open in the metric space $X$. Define $U_n=\left \{ x ∈ U : \text {dist} (x, X\setminus U) >1/n \right \}.$

Take $D_n=U_{n+1}\setminus U_n,\ $so that $U\setminus U_n=\bigcup^{\infty}_{i=n} D_i\ $ and dist $(D_i, D_j ) ≥1/(i + 1)−1/j> 0$ whenever $i + 2 ≤ j.$

Thus, since $\mu^*$ is a $metric$ measure, we have, for all $A\subseteq X \ ;\ $(wlog $\mu^*(A)< \infty),$

$\mu^*(A\cap D_1)+\cdots +\mu^*(A\cap D_{2n-1})= \mu^*(A\cap (D_1\cup\cdots\cup D_{2n-1})\le \mu^*(A),\ $and similarly for the even $1\leq i\le n.$

Therefore,

$\sum_{i=1}^{\infty}\mu^*(A\cap D_i)\le 2\mu^{*}(A)<\infty.$

So now we can say that $\mu^*(A\cap (U\setminus U_n))=\mu^*(A\cap (\bigcup^{\infty}_{i=n} D_i))\le \sum_{i=n}^{\infty}\mu^*(A\cap D_i)\to 0\ $ as $n\to \infty.$

Using again, the fact that $\mu^*$ is a metric outer measure, we compute

$\mu^*(A\cap U_n)+\mu^*(A\setminus U)=\mu^*((A\cap U_n))\cup (A\setminus U))\le \mu^*(A),\ $and finally

$\mu^*(A\cap U)+\mu^*(A\setminus U)\le \mu^*(A\cap U_n)+\mu^*(A\cap (U\setminus U_n))+\mu^*(A\setminus U)\le \mu^*(A)+\mu^*(A\cap (U\setminus U_n))\ $

and it remains only to pass to the limit to see that the Carotheodory condition is satisfied.