Let $P\mathbb{R}$ be the space of probability measures on $\mathbb{R}$. Is the function 
\begin{align*}
P\mathbb{R} &\to \mathbb{N} \cup \{\infty\}\\
\mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\{x\} > 0 \}
\end{align*}
measurable?