I would like to minimize the functional $\mathcal{F}(\mu) = \int x I(F_\mu(x)\leq\tau) d\mu(x)$. However, I'm don't understand how to find the first variation of the term $I(F_\mu(x)\leq\tau) = I(\mu((-\infty,x])\leq\tau)$, with respect to $\mu$.
How do I compute the first variation $\frac{\delta }{\delta \mu}I(\mu((-\infty,x])\leq\tau)$?