Is there a notion of tightness for random elements defined on different probability spaces?

Question

In the case of a sequence of random elements (with values in some metric space $M$) defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$, tightness is defined as follows: $(X_n)_{n\ge 1}$ is tight if for every $\epsilon>0$, there exist a compact set $K\subset M$ such that $\mathbb{P}(X_n\in M\setminus K)<\epsilon$ for all $n\ge 1$.

I was wondering if this can be generalised to when the random elements are not necessarily defined on the same space. I have tried looking around (mainly googling and Durrett: Probability theory and examples) but have not come across such a generalisation. I believe it would look something like the following:

Suppose we have a sequence of probability spaces $(\Omega_n,\mathcal{F}_n,\mathbb{P}_n)$, and random elements $X_n$ on these spaces taking values in some metric space $M$. Then $(X_n)_{n\ge 1}$ is tight if for every $\epsilon>0$, there exist a compact set $K\subset M$ such that $\mathbb{P}_n(X_n\in M\setminus K)<\epsilon$ for all $n\ge 1$.

If this definition is incorrect, please let me know. On the other hand, if you know of a reference which will have what I am looking for, please do tell (or alternatively tell me the correct definition :))

Thanks!

Answer 1

The usual definition of tightness and uniform tightness is that it is a property of a measure or of a set of measures, and not of random elements. For example, Definition 8.6.1 in Bogachev's Measure Theory, vol 2, p.202 (of the Springer English-language edition) starts out

A family $\mathcal M$ of Radon measures on a topological space $X$ is called uniformly tight if for every $\epsilon>0$, there exists a compact set $K_\epsilon$ such that $|\mu|(X\setminus K_\epsilon)<\epsilon$ for all $\mu\in\mathcal M$. A family of Baire measures ...

See also the wikipedia article on this subject for proof that this point of view is widespread.