Talagrand's convex distance inequality ($\mathbb{P}(A)\mathbb{P}(d_T(X,A)\ge s)\le e^{-s^2/4}$)

On a finite product of standard probability spaces, for any measurable event $A$ of positive probability,Consequently $\mathbb{P}(A)\mathbb{P}(d_T(X,A)\ge s)\le e^{-s^2/4}$. Here $d_T$ is the Talagrand convex distance. The distance-event conclusion can be read with outer probability when necessary.