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.

I would like to know the answers to the following two questions.

Let $S$ be a locally compact Hausdorff space, $\mu$ be a regular Borel measure with non-compact support $M$. Denote $$ \mathscr{H}=\{\mathcal{H}\subset 2^M: \mathcal{H}\mbox{ is a disjoint family of Borel sets of positive measure}\} $$ Note that for measures with the continuous part or infinitely many atoms there is an infinite family $\mathcal{H}\in\mathscr{H}$.

Question #1. Does there exist an infinite family $\mathcal{H}\in\mathscr{H}$ such that for any compact $K\subset M$ only finitely many elements of $\mathcal{H}$ have positive measure intersection with $K$?

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.

Maybe it would be easier to characterize measures with the opposite property.

Question #2. What can we say about $S$ or $\mu$ if for any countable $\mathcal{H}\in \mathscr{H}$ there exists a compact $K$ such that infinitely many elements of $\mathcal{H}$ have positive measure intersection with $K$.

It looks like these questions fit well into the realm of measure algebras but I don't know much about them.

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 with $K$?

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.

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.