The following is an overly fancy way of asking a question suggested in Borel Measures: Atoms vs. Point Masses
Let $\phi$ be a property that topological spaces can have (such as "compact", "$T_1$", etc.) and let $M(\phi)$ be the statement "There exists a topological space $X$ with property $\phi$, and a nontrivial, countably additive, 0-1 valued Borel measure on $X$ which is not a point mass." (For the purposes of this question, a measure $\mu$ is a point mass if there exists $a \in X$ such that $\mu(A) = 1$ iff $a \in A$.)
For various values of $\phi$, what is the consistency strength of $M(\phi)$?
A few examples:
$M(\text{compact Hausdorff})$ is a theorem of ZFC (let $X = [0,\omega_1] = \omega_1+1$ with the order topology, and consider the Dieudonné measure.)
$M(\text{finite discrete})$ is inconsistent with ZFC.
$M(\text{separable metrizable})$ is also inconsistent with ZFC, as I show in this answer.
$M(\text{discrete})$ is implied by "there exists a measurable cardinal", and if I understand the Wikipedia article correctly, they are actually equivalent. It's known that this has greater consistency strength than ZFC but is not known to be inconsistent.
I would be particularly interested in knowing about $M(\text{metrizable})$ and $M(\text{completely metrizable})$.
Suppose $X$ is a metric space with a two valued diffused Borel measure. Let $\mathcal{U} = \bigcup \{\mathcal{U}_n : n < \omega\}$ be a basis for $X$ where each $\mathcal{U}_n$ is a disjoint family of open sets - See here.
Let $\langle x_i : i < \kappa \rangle$ list $X$. For each $i$, let $B_i$ be a open ball around $x_i$ whose measure is zero. Let $\theta \leq \kappa$ be least such that $\bigcup \{B_i : i < \theta\}$ is not null.
For each $n$, consider the family $\mathcal{F}_n = \{U \in \mathcal{U}_n :(\exists i < \theta)(U \subseteq B_i)\}$. Each $\mathcal{F}_n$ is a disjoint family of open sets and if $\mathcal{F} = \bigcup \{\mathcal{F}_n : n < \omega\}$, then $\bigcup \mathcal{F}$ is not null. Hence for some $n$, $\bigcup \mathcal{F}_n$ is not null. So we can define a total two valued measure on $|\mathcal{F_n}| \leq \kappa$ hence also on $\kappa$.