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.

The singleton chart integral is ${\mu }_w(\{p\})=w(p){\lambda }_0({\mathbb{R}}^0)=w(p)$. Countable additivity on the disjoint singleton decomposition of any A gives ${\mu }_w(A)={\sum }_{p\in A}w(p)$. The empty sum is zero; zero or infinite weights cause no cancellation.

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. Thus finite positive weights satisfy the positive smooth density theorem and give a Radon measure; directly, compact masses are finite sums, and every set is open and approximated in measure by its finite subsets.

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.