# What Does "Collection of Measure is Tight?'' Mean

I know that a collection of probability measure $$(\mu_\varepsilon )_{\varepsilon > 0}$$ on a (topological) measure space $$(X,\mu)$$ is tight if for all $$\varepsilon >0$$, there is a compact $$K_\varepsilon \subset X$$ such that $$\mu_\varepsilon (K_\varepsilon)>1-\varepsilon$$ for all $$\varepsilon >0$$.

Question : What does this mean concretely, and why is this important ?

Concretely and intuitively, this means that the global behavior of $$(\mu_t )_{t> 0}$$ is not too far away from that over a compact set. And the compactness property help to derive good feature of the measures (for example, the space of continuous functions on a compact set is easier to handle than the space of continuous and bounded functions on the whole set).
• Can we say that if $(\mu_\eta)$ is tight, then, up to a compact set, they are all supported on the same set ? – user659895 Sep 17 at 9:52