Weighted counting in dimension zero

Each point has an open singleton chart, so M is discrete and every subset is Borel. Fix a countable base $(B_j)$. For each $p$ the base contains $\{p\}$; assigning the least such index injects M into $\mathbb{N}$. Thus M is countable without selecting a chart for each point. Singleton indicators form a locally finite smooth partition in dimension zero.

Define ${\mu }_w(A)=\sup \{{\sum }_{p\in F}w(p):F\subseteq A\text{ finite}\}$ for every $A\subseteq M$; the empty sum is zero. This is the displayed nonnegative sum because step 1.1 gives a fixed injection of $M$ into $\mathbb{N}$. If $(A_j)$ is a disjoint sequence, both ${\mu }_w({\bigcup }_j{A}_j)$ and ${\sum }_j{\mu }_w(A_j)$ are the supremum of the same finite weighted subsums: each finite subset of the union meets only finitely many $A_j$, and a finite choice of finite subsets of those $A_j$ has finite union. Hence ${\mu }_w$ is countably additive, so it is a Borel measure by [F2]. Its singleton mass is $w(p){\lambda }_0({\mathbb{R}}^0)=w(p)$, the required chart coefficient integral. Conversely, countable additivity forces any Borel measure with these singleton chart masses to equal ${\mu }_w$ on every subset. No choice of charts or global partition is needed.

A compact subset of a discrete space is finite, since its singleton cover has a finite subcover. Every scalar function here is smooth in local zero-dimensional coordinates. If all weights are finite and positive, compact masses are finite sums. Every subset is open, so it is its own outer-regular open superset. Its measure is the supremum of the masses of its finite subsets by the nonnegative-sum formula, and finite subsets are compact; this proves inner regularity on open sets. Thus the weighted measure is Radon directly, without the general regularity theorem.

For $M=\mathbb{N}$ and $w(k)=2^{-k-1}$, the partial sum through $k=N-1$ is $1-2^{-N}$, which tends to one. For $w(k)=1$ the same partial sum is N, hence the total mass is infinite. In the one-point case the formula gives precisely its weight.