I was reading the wikipedia article on concentration inequality, a topic I am not familiar with.
If one defines a function ($A_{\epsilon}$ is the $\epsilon$-inflation of $A$) $$\alpha(\epsilon)=\sup\{\mu(X\setminus A_{\epsilon})|A\text{ is a Borel set and }\mu(A)\ge 1/2\}\qquad(*),$$ we say that $\mu$ concentrates on the set $A$ if $\alpha(\epsilon)\le Ce^{-c\epsilon^2}$ for some constants $C, c>0$
I was wondering why we choose the sets $A$ to have measure $\mu(A) \ge 1/2$. Is it important that it be $1/2$ in particular and what is the logic behind choosing this $1/2$ number?
Also, it seems to me that even if $\alpha(\epsilon)\le Ce^{-c\epsilon^2}$, the set $A=A(\epsilon)$ returned by the optimisation problem $(*)$ may be different for every $\epsilon$, so in that case on which one of these $A$ does the measure concentrates?