Let \((\Omega, {\mathcal F}, {\mathbb P})\) be a probability space, with \({\mathcal F}\) being the set of all events possible. Recall that the technical term for \({\mathcal F}\) is a sigma-algebra, and it is a set of subsets of \(\Omega\) that includes \(\Omega\), and is closed under complementation and countable unions (and therefore also countable intersections).
In class, we said that a random variable is a function from \(\Omega\to{\mathbb R}\). This is true, all random variables indeed are functions from \(\Omega\to {\mathbb R}\). But not all functions from \(\Omega\to {\mathbb R}\) are random variables, just like not all subsets of \(\Omega\) are allowed to be events, if we are to assign probabilities to events of form \(X \le a\), and so on.
Recall that \({\mathbb R}\), the set of all real numbers is uncountable. Therefore, if were to define a probability space on \({\mathbb R}\), not every subset of \({\mathbb R}\) can be an event. So we first specify which subsets of \({\mathbb R}\) we really want to be events—and those are simple intervals of form \([a,b]\), \((a,b]\), \([a,b)\) and \((a,b)\) for all values of \(a\) and \(b\) (\(-\infty\) and \(\infty\) allowed).
Then we construct a set of all events that includes every interval as above (ie, a set of subsets of \({\mathbb R}\) closed under complementation, countable unions and therefore countable intersections, and which includes \({\mathbb R}\)). This is a sigma-algebra. It includes all the intervals, plus all countable unions and intersections of the intervals. That isn’t enough, because sets such as the Cantor set are complements of countable unions, but are themselves not a countable union of simple intervals. So we have to include many further sets for technical reasons. The smallest collection of subsets of \({\mathbb R}\) that contains all intervals, and in addition, is a sigma-algebra, is called the Borel sigma algebra. We denote this Borel sigma-algebra by \({\mathcal B}\).
For any set \(A\subset {\mathbb R}\), let
\[X^{-1}(A) = \{ \omega\in\Omega: X(\omega) \in A \}\]be the set of all elements of \(\Omega\) that are mapped into \(A\). The notation \(X^{-1}\) is not an inverse function, and in no way implies that the mapping \(X\) is injective or surjective, it is simply a mnemonic to remind ourselves that these are the elements of \(\Omega\) that map into \(A\). In any case, \(X^{-1}(A)\) is a subset of \(\Omega\).
A function \(X:\Omega\to {\mathbb R}\) is defined to be measurable if for all \(B\in {\mathcal B}\), we have \(X^{-1}(B) \in {\mathcal F}\).
Essentially this means that any event in the Borel Sigma-algebra naturally gets a probability under the mapping \(X\), using the probability space \((\Omega, {\mathcal F}, {\mathbb P})\). To do so, for any event \(B\in {\mathcal B}\), we simply identify \(X^{-1}(B)\) and because of measurability, we are assured that \(X^{-1}(B)\in{\mathcal F}\). Now \({\mathbb P}\) has assigned every element in \({\mathcal F}\) a probability, and therefore \(X^{-1}(B)\) in particular, and that is the probability we assign for \(X \in B\).
If \(X\) is not measurable, it means there may be a set \(B\in {\mathcal B}\) such that \(X^{-1}(B)\notin {\mathcal F}\), and therefore cannot be assigned a probability. This means that the regular things we do with manipulation of real numbers, countable unions and complements will break.
A random variable \(X\) is a measurable function \(X:\Omega\to{\mathbb R}\).
You may have wondered why we make up a new notation (random variable) instead of just saying real-valued functions on \(\Omega\). Now you know. Not all real-valued functions on \(\Omega\) are admissible as random variables.
You may have guessed also that there is nothing sacrosanct about real numbers. Indeed, we can replace \({\mathbb R}\) and the Borel sigma-algebra \({\mathcal B}\) by another set \(Z\) and a sigma-algebra on \(Z\). You would then have a \(Z-\)measurable random variable. The real values are chosen in an introductory course since they are simple, ubiquitous, and very very useful.