A sigma-algebra on a set Ω is a collection of subsets closed under countable unions and complementation, serving as the foundation for measure theory. Measurable sets are the elements of a sigma-algebra, forming the domain on which probability measures are defined. Sigma-algebras are essential because probability measures cannot be defined on all subsets, only measurable ones.
Start with finite sigma-algebras generated by simple partitions. Work through examples showing why closure under countable (not just finite) unions is necessary. Examine generated sigma-algebras σ(𝒜).
You know from set theory what a set is, and you know the basic operations: union, intersection, and complement. A sigma-algebra (σ-algebra) is a collection of subsets that is *closed* under these operations — but with one crucial upgrade over ordinary algebra: closure must hold for *countably infinite* unions, not just finite ones. This might seem like a technical tweak, but it's the entire foundation of modern probability theory.
Why do we need this structure at all? The problem is that "probability" is easy to define for simple events — the probability of rolling a 3 is 1/6, and so on — but becomes fragile when we ask about infinite or complicated combinations of events. If probability is a function P assigning numbers to sets of outcomes, we want P to behave consistently: P(A ∪ B) = P(A) + P(B) when A and B don't overlap, and more generally, P(A₁ ∪ A₂ ∪ ...) = ΣP(Aₙ) for disjoint countable collections. For this to make sense, all those unions must be events we can assign probability to — they must be "measurable." The sigma-algebra specifies exactly which sets get to be called events.
Formally, a sigma-algebra 𝒻 on a sample space Ω satisfies three axioms: (1) Ω ∈ 𝒻 (the whole space is measurable), (2) if A ∈ 𝒻 then Aᶜ ∈ 𝒻 (complements are measurable), and (3) if A₁, A₂, ... ∈ 𝒻 then ∪ Aₙ ∈ 𝒻 (countable unions are measurable). From these, you can derive that the empty set is measurable (complement of Ω), countable intersections are measurable (via De Morgan), and any set built from finitely many operations on measurable sets is measurable.
The most important sigma-algebra in practice is the Borel sigma-algebra on ℝ, denoted 𝒻(ℝ). It is the *smallest* sigma-algebra containing all open intervals. This means it contains all open sets, all closed sets, all countable intersections and unions of these — essentially every set you would naturally encounter. Importantly, the Borel sigma-algebra does *not* contain all subsets of ℝ: there exist non-measurable sets (provably, under the axiom of choice), but constructing one requires highly pathological reasoning. In practice, every set you can write down is Borel measurable.
The sigma-algebra is not just a technical detail — it encodes *information*. In probability and stochastic processes, a coarser sigma-algebra (fewer measurable sets) represents less information about outcomes. A finer one represents more. This idea becomes central when you study conditional expectations, filtrations in stochastic processes, and the precise meaning of "what information is available at time t." The sigma-algebra is the mathematical language for saying what events are distinguishable, and therefore what probabilities can be meaningfully defined.