Revision as of 11:15, 8 June 2020

Statement

Theorem.^[1] Let

\mu _{n}

be a sequence of tight probability measures on Polish space

X

. Then, there exists

\mu \in P(X)

and convergent subsequence

\mu _{n_{k}}

such that

\mu _{n_{k}}\rightarrow \mu

in the dual of

C_{b}(X)

. Conversely, every sequence

\mu _{n}\rightarrow \mu

is tight.

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+== Statement ==
+: '''Theorem.'''<ref name=Santambrogio /> Let <math> \mu_{n} </math> be a sequence of tight probability measures on Polish space <math> X </math>. Then, there exists <math> \mu \in P(X) </math> and convergent subsequence <math> \mu_{n_{k}}</math> such that <math> \mu_{n_{k}} \rightarrow \mu </math> in the dual of <math> C_{b}(X) </math>. Conversely, every sequence <math> \mu_{n} \rightarrow \mu </math> is tight.
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