I would like to know the answer to the following question.

Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Does there exist an infinite disjoint family of Borel sets $\mathcal{H}$ such that

• every element of $\mathcal{H}$ has positive measure and sits inside $M$;
• for any compact $K\subset M$ only finitely many elements of $\mathcal{H}$ intersect $K$?

A few side notes:

• I know, how to prove this in the case where $S$ is $\sigma$-compact;
• There is an obvious example without $\sigma$-compactness - counting measure on an uncountable set;
• From [342B, Measure theory. Vol 3. Measure algebras. D. H. Fremlin] I know how to construct at most countable disjoint family $\mathcal{H}$ consisting of compacts of positive measure.