I am reading Santambrogio's book on optimal transport, remark 1.19. Let's consider an optimal transport problem between $(X,\mu)$ and $(Y,\nu)$.

(Remark 1.19) ... every time that we know that any optimal $\gamma$ must be induced by a map $T$, then we have uniqueness (of a Kantorovich solution). Indeed, suppose that two different plans $\gamma_1=\gamma_{T_1}, \gamma_2=\gamma_{T_2}$ are optimal: consider $\frac{1}{2}\gamma_1+\frac{1}{2}\gamma_2$, which is optimal as well by convexity. This last transport plan cannot be induced by a map unless $T_1=T_2$ $\mu-a.e.$, which gives a contradiction.

Here an optimal plan $\gamma$ refers to a solution of the Kantorovich problem and an optimal map $T$ refers to a solution of the Monge problem. The remark does not assume any regularity of the spaces.

I could prove the statement assuming that $\gamma(\{(x,y)\in X\times Y:y=T(x)\})=1$, but I could only show that $\gamma$ is concentrated on the closure of the set. From that I could deduce $\gamma(\{(x,y)\in X\times Y:y=T(x)\})=1$ when $T$ is continuous $\mu$-a.e. I am also not sure if optimality matters for the statement. So my questions are:

1. Can we prove or disprove that $\gamma(\{(x,y)\in X\times Y:y=T(x)\})=1$?
2. If the answer for 1. is no, how can we prove Remark 1.19?