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Measurable sets are elements of a σ-algebra. They satisfy closure under complements and countable unions, allowing rigorous definitions of measure and integration. Understanding measurable set properties is foundational for building measure spaces and extending measures.
From your prerequisite on σ-algebras, you know that a σ-algebra on a set X is a collection ℱ of subsets of X that contains X itself, is closed under complements, and is closed under countable unions. The elements of ℱ are called measurable sets. The key question to ask now is: why these closure properties specifically? What goes wrong without them?
The answer lies in what we want to do with measurable sets: assign them a "size" (a measure) that behaves consistently. If a set A is measurable, its complement Aᶜ should also be measurable — otherwise we could measure "everything outside A" but not A itself, which is incoherent. Countable unions are essential because measure theory is built around limits: the measure of a countable union of disjoint sets should equal the sum of their individual measures (countable additivity). To state this axiom, all those unions must be measurable in the first place. Finite collections would not suffice for analysis, where limits of sequences of sets arise constantly.
The closure properties generate derived properties automatically. Countable intersections are measurable, because ∩Aₙ = (∪Aₙᶜ)ᶜ — complement the union of complements. Set differences are measurable: A \ B = A ∩ Bᶜ. The empty set is measurable: ∅ = Xᶜ. Symmetric differences, limsups, and liminfs of sequences of measurable sets are all measurable. This algebraic richness means that any set you construct from measurable sets through the operations of analysis remains measurable — it never "falls out" of the σ-algebra unexpectedly.
The classic example is the Borel σ-algebra on ℝ, generated by the open intervals. Every open set, closed set, Fσ set, Gδ set, and their countable combinations are Borel measurable. The Lebesgue σ-algebra is larger, adding completions — null sets and their subsets. The non-measurable sets that analysts construct (like Vitali sets) require the axiom of choice and are explicitly excluded from any σ-algebra by their construction. The σ-algebra structure is precisely what separates the sets we can measure from those we cannot, and the closure properties are the mechanism that keeps the "measurable" world self-consistent under the operations of analysis.