Why mathematicians cannot give every set a size, and what they do instead
Try to assign a length to every possible subset of the real line and you hit a wall: the axiom of choice guarantees sets, such as the Vitali sets, that have no sensible length at all. The sigma-algebra is the workaround, a carefully chosen family of subsets that can be measured.
Analysts and probability theorists lean on the sigma-algebra whenever they need to say which sets have a well-defined size. In calculus that size might be an area or a volume; in probability it is the chance of an event. The Greek sigma in the name echoes the German word Summe, meaning sum, and the structure is sometimes called a sigma-field.
Three closure rules define it, and each mirrors common sense about size. If an event has a probability, then its failure to happen should too, so the family must contain complements. If several sets each have a size, their combination should as well, so it must survive countable unions. And if individual events have probabilities, so should the event that all of them occur together, which means countable intersections are included. A topology looks similar but only needs finite intersections, and a plain set algebra is closed under finite operations alone.
Examples range from trivial to deep. Any finite algebra of sets automatically qualifies. Take any partition of a set into countably many pieces, and the unions of those pieces form one. The important case on the real line starts from open intervals and keeps adding countable unions, intersections and complements, iterating through the countable ordinals until nothing new appears; that construction is known as the Borel hierarchy.
Mathematicians give three main reasons for the concept. It underpins measures, which want the size of disjoint pieces to add up even across infinitely many of them. It lets them take limits of sequences of sets, vital for ideas like almost sure convergence. And a smaller sigma-algebra sitting inside a larger one can represent partial information, which matters for conditional expectation; a sequence of coin flips, the Bernoulli process, is the standard illustration.
Source: Σ-algebra
