Is this a misuse of the term "probability space"? Let me first state the definitions as I am using them.
Do correct me if I am wrong here!


*

*A "probability space" is a triple $(\Omega, F \subseteq 2^{\Omega}, \mu : F \rightarrow [0,1])$. The intuition being that the $\Omega$ is the set of possible "outcomes" and $F$ is the set of "events" whereby an event is a set of outcomes and $\mu$, the "probability measure", is a map that assigns a probability $\mu(E)$ for every event $E  \in F$.


*A random variable $X$ is a function that maps $X : \Omega \rightarrow [0,1]$ such that $\sum_{\omega \in \Omega} X(\omega) = 1$


*And given a random variable it automatically defines a $\mu : F \rightarrow [0,1]$ as $\mu(E) = \sum_{e \in E} X(e)$ for all $E \in F$


*

*Hence is the triple $(\Omega, F \subseteq 2^{\Omega}, \mu : E \rightarrow [0,1])$ is the same information as $(\Omega, X : \Omega \rightarrow [0,1],  F \subseteq 2^{\Omega} )$ ?


*So in the discrete case (of say $n$ events) is defining a "probability distribution" $P = (p_1,p_2,..,p_n)$ the same as defining a "random variable" ? They are just two words for the same thing i.e "probability distribution = random variable" ?


*Typically a "probability space" as defined as the triple above seems to come with the notion of a probability measure $\mu$. But then in certain contexts like say while defining a Renyi divergence $D_\alpha (P \vert \vert Q)$, I see use of the phrase "Let $P$ and $Q$ be defined on the same probability space". This doesn't make sense to me.
Is this a misnomer? As in what they really want to say is is "Let $P$ and $Q$ be two different probability distributions on the same set of outcomes" ?  So basically they have $2$ different "probability spaces", $(\Omega, P : \Omega \rightarrow [0,1],  F \subseteq 2^{\Omega} )$ and $(\Omega, Q : \Omega \rightarrow [0,1],  F \subseteq 2^{\Omega} )$ ?
 A: Let us assume that $\Omega$ is finite or countable.
I think the main confusion here is the relationship between a probability measure and a random variable. A probability measure $\mathbb P$ is a function $\mathbb P:\mathcal F \rightarrow [0, 1]$ for some $\sigma$-algebra. A random variable, on the other hand, is a function $X: \Omega \rightarrow S$ where $S$ is some set, not $[0, 1]$ in general. I.e. it maps outcomes to some other values, whereas the probability measure assigns probabilities to events in the $\sigma$-algebra.
The distribution:
You are correct in thinking that a random variable can be characterised by its distribution, but the two things are not the same. The distribution assigns probabilities to the events that $X$ takes certain values. The distribution $P^X$ of $X$ is defined by
$$
P^X(A) = \mathbb P(\lbrace \omega: X(\omega) \in A \rbrace)
$$
for $A \in S$. Let $S'$ be the image of $\Omega$ under $X$. Indeed, the distribution of $X$ defines a probability measure on $(S', 2^{S'})$, but not on the original space $(\Omega, 2^{\Omega})$. 
So you see, the distribution of $X$ defines a probability measure on some other space than $(\Omega, 2^{\Omega})$. Meanwhile you can define alternative probability measures on the measurable space $(\Omega, 2^{\Omega})$ as much as you want.
So when they say "Let $\mathbb Q$ and $\mathbb P$ be defined one the same probability space" they mean two different probability measure on the measurable space $(\Omega, 2^{\Omega})$.
This is of course not an exhaustive treatment of the problem, but hope it serves to clarify some parts of it. 
