the Borel sigma-algebra
/ boh-REL /
Once you accept that not every subset of the real line can be an event, you need a concrete, standard family of sets to work with — one that contains all the sets you actually care about (intervals, points, and everything you can build from them) but stops short of the pathological ones. The Borel sigma-algebra is that family on the real line. It is the natural home for continuous random variables, and almost any set you can describe in ordinary language turns out to be Borel.
It is defined as the smallest sigma-algebra that contains all the open intervals. The phrase smallest sigma-algebra containing a collection means: throw in those intervals, then add whatever you are forced to add to satisfy the three sigma-algebra rules — complements and countable unions — and nothing more. Starting from intervals like (a, b) and closing up under those operations, you generate every open set, every closed set, every single point, the rationals, half-lines, countable unions of intervals, and far more. You could equally start from all open sets, or all half-lines (-infinity, x], and reach the same family. The Borel sets are vast, yet they form a strict subfamily of all subsets: the Vitali set is not Borel.
Why this exact choice? Because it is the smallest family that makes every continuous function (and hence every reasonable random variable) measurable, while still excluding non-measurable sets. When you define a probability on the line by giving a cumulative distribution function, you are really defining a measure on the Borel sets. The half-line (-infinity, x] is the key building block: a function is Borel-measurable precisely when the preimage of each such half-line is measurable, which is why distributions are pinned down by F(x) = P(X <= x).
The set of all rational numbers in [0, 1] is Borel: it is a countable union of single points, and each point {q} is closed, hence Borel, so the union is too. Its Lebesgue measure is 0, even though the rationals are dense in the interval.
Built from intervals by countable unions and complements, the Borel sets capture everything describable while excluding the exotic.
Borel sets are not quite all measurable sets: the complete Lebesgue sigma-algebra is a bit larger (it also includes every subset of a measure-zero set), but for nearly all of probability the Borel sets are exactly what you need.