A probability space consists of the following data:
The sample space $S$, which is the set of possible outcomes (of an experiment.
The event algebra $A$, where each event consists of a set of outcomes in $S$, and the collection of events constitutes a $\sigma$-algebra – it is closed under countable sequences of union, intersection and complement operations (and hence set differences). Implied here is that the empty set and whole sample space are events in $A$.
A measure function $P$, which assigns a probability to each event in $A$. $P$ must be additive on countable disjoint unions, and must assign 1 to the whole sample space $S$.
A random variable is a function $X$ from the sample space S into a range space $V$, which is measurable, which means: there is a $\sigma$-algebra of subsets of $V$, and the inverse image of every such subset under the function $X$ is an event in $A$.