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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 ?

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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).

The point is that tight families on a metric space play the same role as relatively compact sets in general topology. In particular, one can extract from a tight family a convergent subsequence (Prokhorov's theorem). This is fundamental when one wants to prove the convergence in distribution because usually the strategy is to prove tightness and prove that the potential limit of a subsequence is unique.

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  • $\begingroup$ Can we say that if $(\mu_\eta)$ is tight, then, up to a compact set, they are all supported on the same set ? $\endgroup$ – user659895 Sep 17 at 9:52

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