Frechet-Hausdorff theorem reference from J.L. Kelley used in proof that each probability measure is inner regular

Question

Theorem: If $S$ is a complete, separable metric space, then each probability measure on it is inner regular.

Proof: Since $S$ is separable, for each $n \in \mathbb{N}$ there exist countably many balls of radius $1/n$ such that $S = \bigcup\limits_{j = 1}^\infty {{K_{j,n}}} = \bigcup\limits_{j = 1}^\infty {{{\overline K }_{j,n}}} $. Let $\varepsilon > 0$. Since $\mathbb{P}$ is continuous with respect to a rising sequence of events, for each $n \in \mathbb{N}$ there exists ${k_n} \in \mathbb{N}$ such that $\mathbb{P}\left( {\bigcup\limits_{j = 1}^{{k_n}} {{{\bar K}_{j,n}}} } \right) > 1 - \frac{\varepsilon }{{{2^n}}}$.

Now comes the part which is unclear to me:

Define ${K_\varepsilon } = \bigcap\limits_{n = 1}^\infty {\left( {\bigcup\limits_{j = 1}^{{k_n}} {{{\overline K }_{j,n}}} } \right)} $; ${K_\varepsilon }$ is a closed set (obviously) and for each $n \in \mathbb{N}$ we have ${K_\varepsilon } \subseteq \bigcup\limits_{j = 1}^{{k_n}} {{{\overline K }_{j,n}}} $, by applying Frechet-Hausdorff theorem(J.L.Kelley, no page reference) it follows that ${K_\varepsilon }$ is compact.

Furthermore, $1 - \mathbb{P}\left( {{K_\varepsilon }} \right) = \mathbb{P}\left( {K_\varepsilon ^C} \right)\sum\limits_{n = 1}^\infty {\mathbb{P}\left( {{{\left( {\bigcup\limits_{j = 1}^{{k_n}} {{{\bar K}_{j,n}}} } \right)}^C}} \right)} < \sum\limits_{n = 1}^\infty {\frac{\varepsilon }{{{2^n}}}} = \varepsilon $ is inner regular.

My question: How do I conclude that ${K_\varepsilon }$ is compact? What does the referenced theorem state?

EDIT: For definition of inner-regularity, see here.

Answer 1

Since $K_\epsilon$ is closed it is complete (as a subspace of $S$).

Choose some $n$. Then the ${{{\overline K }_{j,n}}}$ ($j=1,...,k_n$) cover $K_\epsilon$. Select $x_j \in K_\epsilon \cap {{{\overline K }_{j,n}}}$. Then the $x_j$ form a ${2 \over n}$-net for $K_\epsilon$. Hence for any $\delta>0$ we can find a finite $\delta$-net for $K_\epsilon$ and so it is totally bounded.

Hence $K_\epsilon$ is compact.

Answer 2

I think that it is an "historical" reference; see :

• Nicolas Bourbaki, Elements of the History of Mathematics (French ed 1984), Ch.16 : Metric spaces, page 165 :

the notion of metric space was introduced in 1906 by M.Fréchet, and developed a few years later by F. Hausdorff in his "Mengenlehre".

See :

• Maurice Fréchet (2 September 1878 – 4 June 1973) :

His first major work was his outstanding PhD thesis that he submitted in 1906. The thesis was titled Sur quelques points du calcul fonctionnel and was concerned with the calculus of functionals.

It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term already in his 1904 paper which led to the famous 1906 thesis).

and :

• Felix Hausdorff (November 8, 1868 – January 26, 1942) :

In the summer of 1912 he also began work on his magnum opus, the book "Principles of set theory" [...] released in April 1914 [Grundzüge der Mengenlehre, English ed : Set theory, New York 1957]. Hausdorff's work was the first textbook which presented all of set theory in this broad sense, systematically and with full proofs.

Among the publications of Hausdorff [...] the work Dimension and outer measure from 1919 is particularly outstanding. It has remained highly topical and in later years has been probably the most cited mathematical original work from the decade from 1910 to 1920. In this work, the concepts were introduced which are now known as Hausdorff measure and the Hausdorff dimension.

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The "missing" link can be found - I think - into :

• John Kelley, General topology (1975), page 138 :

THEOREM. If $X$ is a topological space [...] and satisfy the second axiom of countability [see page 48 : a space whose topology has a countable base], then all four conditions are equivalent :

(a) every infinite subset of $X$ has an $\omega$-accumulation point [see page 137 : a point $x$ is an $\omega$-accumulation point of a set $A$ iff each neughborhood of $x$ contains infinitely many points of $A$]

[...]

(c) The space $X$ is compact.