Let $\mu$ be a probability measure supported on $\mathbb R$.
My question is the following:
For a given $\varepsilon \in (0,1)$, does there exist a finite number $F > 0$ such that $\mu([-F, F]) >= 1-\epsilon$ holds? In other words, does a probability measure assign 'almost all of the mass' to a finite interval always?
If the answer is 'no', then do we have a name for the measures that indeed satisfy this?
The answer is yes since $\{[-F;F] \}_{F \in \Bbb N }$ are an increasing sequence of intervals hence $$\lim_{F \to \infty} \mu ( [-F; F]) = \mu ( \lim_{F \to \infty} [-F; F] ) = \mu ( \Bbb R ) = 1$$