Let $\mu$ be the Lebesgue measure on $[0, 1]$ and $\nu$ be the Borel measure defined on $[0, 1]$ by $$\int_{[0, 1]} f(y) \nu(dy) = (1 - \alpha) \int_{0}^{1} f(y) dy + \alpha f(1) ~~~~~ \forall f \in \mathcal{C}({\mathbb{R}})$$ Where, $\alpha$ is fixed in $[0,1]$. What is the optimal transport map $T$ sending $\mu$ to $\nu$?
May someone provide me with some hints about this question? please!