The situation: Let $X$ and $Y$ are Polish spaces equipped with Borel probability measures $\mu$ and $\nu$ such that $\nu=h_*\mu$ is the push forward measure with regard to a continuous (measurable) function $h:X\to Y$.
It is commonly known that in this case $\mu_y(\cdot)=\mu(\cdot\vert h=y)$ is a regular conditional probability with regard to $h$. So $$ \int_A \mu_y(\cdot)d\nu=\mu(\cdot\cap h^{-1}(A)) $$ It is also known that for the fiber measures $\mu_y(h^{-1}(\{y\}))=1$. Assume that we also know the support of the fiber measures $supp(\mu_y)$ is finitely bounded( $\#supp(\mu_y)<n$). Obviously $\mu_y$ is the sum of weighted dirac measures in this case and $supp(\mu_y)\subseteq h^{-1}(\{y\})$.
Are there any conditions on anything such that we can force $supp(\mu_y)= h^{-1}(\{y\})$ $\nu$-almost surely? Do we have any additional properties of the fiber measures?