Understanding tightness of a set of measures (any subset of a tight set is tight)
By Emma Valentine •
This is the definition I have of tightness of a set of measures on a metric space:
Let $(X, d)$ be a metric space. A family of probability measures $\Pi$ defined on the Borel sets of $X$ is said to be tight if for each $\epsilon > 0$, there exists a compact set $K_{\epsilon} \subset X$ such that for all $\mu \in \Pi$, $\mu(K_{\epsilon}) > 1 - \epsilon$.
Does this imply that any subset of a tight space is tight? Surely we can just use the same sets $K_{\epsilon}$ to verify the tightness condition. This feels to good to be true though.
$\endgroup$ 4 Reset to default
