I am trying to understand measure concentration on a unit sphere, using this text book.
Lectures on Discrete Geometry by Jiří Matoušek https://link.springer.com/book/10.1007/978-1-4613-0039-7
On page 331, there is a theorem (Measure concentration for the sphere) as follows.
Let $A\subseteq S^{n-1}$ be a measurable set with $P[A]\geq 1/2$, where $P[A]$ is the probability that a random point falls in $A$. Let $A_{t}$ denote the $t$-neighborhood of $A$, that is, the set of all $x\in S^{n-1}$ whose Euclidian distance to $A$ is at most $t$. Then, $$1-P[A_{t}] \leq 2\exp[-t^{2}n/2].$$ Thus, if A occupies half of the sphere, almost all points the sphere lie at distance at most $\mathcal{O}(n^{-1/2})$.
I can't understand what the last sentence means. If $t=n^{-1/2}$, I can see that the right-hand side of the inequality becomes, $$2\exp[-1/2]=1.21$$ but how can I use this? Thank you.