Proof of Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight

Each result cited below is universally quantified over the data in its own statement and is applied with the data named at the point of citation.

Let $\mathcal{B}(X)$ be the Borel $\sigma$-algebra of $(X,d)$, on which $\mu$ is a measure by Borel Measure on a Metric Space, and let $\varepsilon$ be a positive real number. By Tight Family of Borel Measures on a Metric Space §tight we must find a set $K\subseteq X$, compact in $(X,\mathcal{T}_{d})$, with $\mu(X\setminus K)\le\varepsilon$.

Step 1 (the empty space). Suppose $X=\varnothing$ and put $K=\varnothing$. The set $K$ is closed in $(X,\mathcal{T}_{d})$ by claim 1 of Complements, Unions and Intersections of Closed Sets in a Topological Space, and it is totally bounded in $(X,d)$ in the sense of Totally Bounded Subset of a Metric Space, the empty finite set $F=\varnothing$ serving for every radius. Hence $K$ is compact in $(X,\mathcal{T}_{d})$ by A Closed Totally Bounded Subset of a Complete Metric Space is Compact §compact, applied to the complete metric space $(X,d)$ and the set $K$, and $\mu(X\setminus K)=\mu(\varnothing)=0\le\varepsilon$ by Measure, Measure Space, and Probability Measure. From now on we assume $X\neq\varnothing$.

Step 2 (an enumerated dense set, and coverings by closed balls). By Separable Metric Space there is a countable set $D\subseteq X$ that is dense in $X$ for $\mathcal{T}_{d}$, that is, $\operatorname{cl}_{X}(D)=X$ by Dense Subset of a Topological Space. Let $x\in X$ and $r$ a positive real number. Since $x\in\operatorname{cl}_{X}(D)$, the implication from condition 1 to condition 3 of Characterization of the Closure in a Metric Space by Open Balls gives $a\in D$ with $d(x,a)<r$. Taking any $x\in X$ shows that $D$ is nonempty, so by Countable Set there is a sequence $(q_{i})_{i\in\mathbb{N}}$ in $D$ such that every point of $D$ equals $q_{i}$ for some $i\in\mathbb{N}$. Consequently, for every $x\in X$ and every $m\in\mathbb{N}$ there is $i\in\mathbb{N}$ with $d(q_{i},x)=d(x,q_{i})<\tfrac{1}{m}$, using the symmetry condition 3 of Metric Space; that is, $x$ lies in the closed ball $\bar{B}_{d}(q_{i},\tfrac{1}{m})=\{y\in X:d(q_{i},y)\le\tfrac{1}{m}\}$ of Closed Ball in a Metric Space.

Step 3 (finite unions of closed balls). For $m,N\in\mathbb{N}$ let $C_{m,N}=\bigcup_{i=1}^{N}\bar{B}_{d}(q_{i},\tfrac{1}{m})$. Each of these closed balls is closed in $(X,\mathcal{T}_{d})$ by claim 3 of Elementary Properties of the Closed Ball in a Metric Space, so $C_{m,N}$ is closed in $(X,\mathcal{T}_{d})$ by claim 2 of Complements, Unions and Intersections of Closed Sets in a Topological Space, and $C_{m,N}\in\mathcal{B}(X)$ by claim 1 of Borel Measurability and Bounded Integration on a Metric Space. For fixed $m$ we have $C_{m,N}\subseteq C_{m,N+1}$ for every $N$, and $\bigcup_{N\in\mathbb{N}}C_{m,N}=X$ by Step 2.

Step 4 (choice of the radii counts). Fix $m\in\mathbb{N}$. By the monotonicity Basic Properties of a Measure §monotone, $\mu(C_{m,N})\le\mu(X)$ for every $N$, and $\mu(X)$ is real by hypothesis; so every $\mu(C_{m,N})$ is real and the set of these values is bounded above by $\mu(X)$. By the continuity from below Basic Properties of a Measure §continuity-below, applied to the increasing sequence $(C_{m,N})_{N\in\mathbb{N}}$ with union $X$, the sequence $(\mu(C_{m,N}))_{N\in\mathbb{N}}$ converges to $\mu(X)$. The number $\varepsilon\bigl(\tfrac{1}{2}\bigr)^{m}$ is positive, so by Limit of a Sequence of Real Numbers there is $N_{m}\in\mathbb{N}$ with $\mu(X)-\mu(C_{m,N_{m}})<\varepsilon\bigl(\tfrac{1}{2}\bigr)^{m}$. Put $C_{m}=C_{m,N_{m}}$. By Basic Properties of a Measure §differences,

$$\mu(X\setminus C_{m})=\mu(X)-\mu(C_{m})<\varepsilon\bigl(\tfrac{1}{2}\bigr)^{m}.$$

The numbers $N_{m}$ are chosen for all $m\in\mathbb{N}$ after $\varepsilon$ has been fixed.

Step 5 (the compact set). Let $K=\bigcap_{m\in\mathbb{N}}C_{m}$. It is closed in $(X,\mathcal{T}_{d})$ by claim 3 of Complements, Unions and Intersections of Closed Sets in a Topological Space, applied to the family $(C_{m})_{m\in\mathbb{N}}$ of closed sets. It is totally bounded in $(X,d)$: given a positive real $\eta$, claim 3 of The Archimedean Property of the Real Numbers gives $m\in\mathbb{N}$ with $\tfrac{1}{m}<\eta$; the set $F=\{q_{1},\dots,q_{N_{m}}\}$ is finite, and every $y\in K$ lies in $C_{m}$, hence satisfies $d(q_{i},y)\le\tfrac{1}{m}<\eta$ for some $i\le N_{m}$, so that $K\subseteq\bigcup_{a\in F}B_{d}(a,\eta)$ with the open balls of Open Ball in a Metric Space, as Totally Bounded Subset of a Metric Space requires. Since $(X,d)$ is complete, A Closed Totally Bounded Subset of a Complete Metric Space is Compact §compact, applied to $(X,d)$ and the closed totally bounded set $K$, shows that $K$ is compact in $(X,\mathcal{T}_{d})$.

Step 6 (the measure of the complement). We have $X\setminus K=\bigcup_{m\in\mathbb{N}}(X\setminus C_{m})$ with every $X\setminus C_{m}\in\mathcal{B}(X)$, so the countable subadditivity Basic Properties of a Measure §subadditivity gives

$$\mu(X\setminus K)\le\sum_{m\in\mathbb{N}}\mu(X\setminus C_{m}),$$

the sum being that of Measure, Measure Space, and Probability Measure. Its terms $a_{m}=\mu(X\setminus C_{m})$ are real, with $0\le a_{m}<\varepsilon\bigl(\tfrac{1}{2}\bigr)^{m}$ by Step 4. For every $n\in\mathbb{N}$, the geometric sum formula of Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, read with $r=\tfrac{1}{2}$, gives $\sum_{m=1}^{n}\bigl(\tfrac{1}{2}\bigr)^{m}=1-\bigl(\tfrac{1}{2}\bigr)^{n}\le1$, hence

$$\sum_{m=1}^{n}a_{m}\le\varepsilon\sum_{m=1}^{n}\bigl(\tfrac{1}{2}\bigr)^{m}\le\varepsilon .$$

Thus the partial sums are bounded above by $\varepsilon$, so by Measure, Measure Space, and Probability Measure the sum $\sum_{m\in\mathbb{N}}a_{m}$ is their least upper bound, which is at most $\varepsilon$. Therefore $\mu(X\setminus K)\le\varepsilon$, with $K$ compact in $(X,\mathcal{T}_{d})$. As $\varepsilon$ was an arbitrary positive real number, the set $\{\mu\}$ is tight in $(X,d)$ by Tight Family of Borel Measures on a Metric Space §tight.