Tightness of measures

If ${\displaystyle X}$ is a metrisable compact space, then every collection of (possibly complex) measures on ${\displaystyle X}$ is tight. This is not necessarily so for non-metrisable compact spaces. If we take ${\displaystyle [0,\omega _{1}]}$ with its order topology, then there exists a measure ${\displaystyle \mu }$ on it that is not inner regular. Therefore, the singleton ${\displaystyle \{\mu \}}$ is not tight.

If ${\displaystyle X}$ is a compact Polish space, then every probability measure on ${\displaystyle X}$ is tight. Furthermore, by Prokhorov's theorem, a collection of probability measures on ${\displaystyle X}$ is tight if and only if it is precompact in the topology of weak convergence.

Consider the real line ${\displaystyle \mathbb {R} }$ with its usual Borel topology. Let ${\displaystyle \delta _{x}}$ denote the Dirac measure, a unit mass at the point ${\displaystyle x}$ in ${\displaystyle \mathbb {R} }$. The collection

${\displaystyle M_{1}:=\{\delta _{n}|n\in \mathbb {N} \}}$

is not tight, since the compact subsets of ${\displaystyle \mathbb {R} }$ are precisely the closed and bounded subsets, and any such set, since it is bounded, has ${\displaystyle \delta _{n}}$-measure zero for large enough ${\displaystyle n}$. On the other hand, the collection

${\displaystyle M_{2}:=\{\delta _{1/n}|n\in \mathbb {N} \}}$

is tight: the compact interval ${\displaystyle [0,1]}$ will work as ${\displaystyle K_{\varepsilon }}$ for any ${\displaystyle \varepsilon >0}$. In general, a collection of Dirac delta measures on ${\displaystyle \mathbb {R} ^{n}}$ is tight if, and only if, the collection of their supports is bounded.

Consider ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ with its usual Borel topology and σ-algebra. Consider a collection of Gaussian measures

${\displaystyle \Gamma =\{\gamma _{i}|i\in I\},}$

where the measure ${\displaystyle \gamma _{i}}$ has expected value (mean) ${\displaystyle m_{i}\in \mathbb {R} ^{n}}$ and covariance matrix ${\displaystyle C_{i}\in \mathbb {R} ^{n\times n}}$. Then the collection ${\displaystyle \Gamma }$ is tight if, and only if, the collections ${\displaystyle \{m_{i}|i\in I\}\subseteq \mathbb {R} ^{n}}$ and ${\displaystyle \{C_{i}|i\in I\}\subseteq \mathbb {R} ^{n\times n}}$ are both bounded.