# Properties of the space of random variables for different types of convergence

Suppose we have a probability space $(\Omega,\mathcal{A},\mathbb{P})$, a sequence of random variable $(X_n)_n$ and some extra random variable $X$ on $(\Omega,\mathcal{A},\mathbb{P})$. There are a lot of different types of convergence which are defined now, for example we have:

• Weak convergence ($X_n\overset{w}{\rightarrow} X$)
• Convergence in probability ($X_n\overset{p}{\rightarrow} X$)
• Almost sure convergence ($X_n\overset{a.s.}{\rightarrow} X$)
• Convergence in $L^p$ ($X_n\overset{L_p}{\rightarrow} X$)
• Sure convergence ($X_n\overset{s}{\rightarrow} X$)
• Convergence in mean ($X_n\overset{m}{\rightarrow} X$)

and for all of these we can look at the space of random variables with this type of convergence $(\mathcal{R}(\Omega,\mathcal{A},\mathbb{P}),\overset{*}{\rightarrow})$ where we sometimes make restrictions like in $L_p$ convergence we only take variables with a finite $p$'th moment. These spaces have certain properties like completeness, compactness, metrizability,$\dots$

My question is if there is some comprehensive list of all of these spaces and which properties they do/do not have? I have seen some properties (like completeness of the $L^p$ space) but a list would be really nice in my opinion. For clarification : I am not looking for properties like "If $X_n$ converges a.s. to $X$ and $Y_n$ converges a.s. to $Y$ then $X_nY_n$ converges a.s. to $XY$, I'm looking for the classical topological/metric properties of the spaces of random variables.

• I'm afraid this question is a bit too broad; a comprehensive answer would start to look like a Wikipedia article. Can you be a bit more specific about what you're asking? – Math1000 Nov 28 '16 at 2:51
• Oke, you might be right, though some reference for a book/wikipedia page where these spaces are looked at in this topological way would also suffice as an answer. Let's say I'm interested in metrizeability, completeness and sequential compactness properties. – HolyMonk Nov 28 '16 at 7:54