Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\mathbb{D}^n)} \quad u(z) = \int u d\mu$$
where $A(\mathbb{D}^n)$ is the polydisc algebra.
We say that $\nu$ is singular with respect to $\mu$, if there are such two disjoint sets $A, B \subset \mathbb{D}^n$ that $A \cup B = \mathbb{D}^n$ and $\nu(A) = 0$, $\mu(B) = 0$. We denote $\nu \perp \mu$.
A null set is a non-zero set $E$ with measure zero. In this case, because we are in the $n$ dimensional case, we will be talking of $E := E_1 \times \ldots \times E_n$, where $E_i$ is a null-set each.
My questions:
Is it always the case, that a measure $\nu$ singular with respect to all representing measures $\mu$ for some point $z \in \mathbb{D}^n$ is always concentrated on $E$?
We know, that $\mu$ will be a Henkin measure. But what about $\nu$? Will it also be a Henkin measure?
I tried finding some information about this topic, but unfortunately, I wasn't able to find any papers which could be useful for me, even after searching for hours. I really hope someone will be able to help me here.
