For $\mu \in \mathcal P(\mathbb R)$, let $m(\mu)$ be the median of $\mu$, defined as the smallest of all medians of $\mu$ as follows: $$ m(\mu) = \inf \left\{ x \in \mathbb R \,\middle|\, \mu((-\infty,x]) \ge \frac12 \right\}. $$
Assume that $\mathcal P(\mathbb R)$ is endowed with the weak convergence topology. Then, $m$ is not continuous as, for exemple, $\mu_n = (\frac12-\frac 1 n) \delta_0 + (\frac12 + \frac 1 n) \delta_1$ weakly converges to $\mu = \frac12(\delta_0+\delta_1)$, but $m(\mu_n)=1$ for all n while $m(\mu)=0$. (Counter exemple given in a comment, thanks)
Now, endow $\mathcal P(\mathbb R)$ with the Borel sigma algebra associated to the above topology. Is $m$ measurable?