Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].

Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume that $\mathcal{S}$ is an element of the $\sigma$-algebra on the space of probability measures on $(X,\Sigma)$ generated by the evaluation mappings $\{\mu \mapsto \mu(A) : A \in \Sigma\}$.]

The set of measures $\mathcal{S}$ is said to be **mutually singular** if for any distinct $\mu_1,\mu_2 \in \mathcal{S}$, the measures $\mu_1$ and $\mu_2$ are mutually singular. So, for example, the set of Borel measures on $[0,1]$ given by
$$ \mathcal{S} \ := \ \{\delta_x : x \in [0,1]\} \cup \{\mathrm{Lebesgue}\} $$
is mutually singular.

Now I can think of some (seemingly) highly natural ways of strengthening the definition of a "mutually singular" collection of measures, such that the above example no longer qualifies:

**Definition.** We say that $\mathcal{S}$ is *A-strongly mutually singular* if there exists a function $L \colon \mathcal{S} \to \Sigma$ such that for any distinct $\mu_1,\mu_2 \in \mathcal{S}$, $\mu_1(L(\mu_1))=1$ and $\mu_1(L(\mu_2))=0$.

**Definition.** We say that $\mathcal{S}$ is *measurably A-strongly mutually singular* if there exists a function $L \colon \mathcal{S} \to \Sigma$ as in the previous definition, with the additional property that the set $\{(\mu,x):x \in L(\mu)\}$ is a measurable subset of $\mathcal{S} \times X$ (where $\mathcal{S}$ is equipped with the $\sigma$-algebra generated by the evaluation maps $\{\mu \mapsto \mu(A) : A \in \Sigma\}$).

**Definition.** We say that $\mathcal{S}$ is *B-strongly mutually singular* if for any two mutually singular probability measures $Q_1$ and $Q_2$ on $\mathcal{S}$ (equipped with the $\sigma$-algebra generated by $\{\mu \mapsto \mu(A) : A \in \Sigma\}$), the probability measures
$$ A \mapsto \int_\mathcal{S} \mu(A) \, Q_1(d\mu) \hspace{5mm} \textrm{and} \hspace{5mm} A \mapsto \int_\mathcal{S} \mu(A) \, Q_2(d\mu) $$
on $(X,\Sigma)$ are mutually singular.

[I suspect that under the standardness assumptions on $X$ and $\mathcal{S}$, being measurably A-strongly mutually singular implies being B-strongly mutually singular; but I haven't yet managed to prove it.]

**My questions:**

Have any of the above (or similar) stronger forms of "mutually singular" been studied before? Do they have standard names?

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a measurable map admitting uncountably many ergodic probability measures, and let $\mathcal{S}$ be the set of ergodic probability measures of $f$. Is $\mathcal{S}$ necessarily "strongly" mutually singular under any of the above (or similar) definitions?