Let $(\Omega, \mathcal F, P)$ be a measurable space (or a probability space) and $A$ a set of real valued Borel-mesurable functions.
What does it mean for $A$ to be bounded in measure (or in probability) ?
2026-04-23 02:09:34.1776910174
What is a set of functions bounded in measure?
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Being bounded in probability, or being tight means that for every $\epsilon>0$, there exists a positive real number $M$ such $\mu_{X_n}[-M,+M]>1-\epsilon$ for all $n$. Here, $\mu_{X_n}$ refers to the law of the random variable $X_n$. Basically, for a set of tight random variables, the mass does not escape to infinity. As an example if our random variables have the distributions $X_n \sim\text{unif}[n,n+1]$, then they are not tight because all the mass escapes to $+\infty$.