Measure Theory 2 | Borel Sigma Algebras [dark version]

A key takeaway is that the “smallest” sigma algebra containing a chosen collection of sets can be built systematically: take all countable intersections of sigma algebras that already contain the collection. This guarantees the result is again a sigma algebra, and it gives a principled way to define measurable sets without guessing.

Sigma algebras are families of subsets of a set X closed under three operations: they always include the empty set and X, they are closed under complements, and they are closed under countable unions. Starting from many candidate sigma algebras on the same X, intersecting them produces a new sigma algebra—no matter how many are intersected, even if the index set is uncountable. That intersection property becomes a construction tool: if a collection M of subsets of X does not yet form a sigma algebra, one can define σ(M) as the smallest sigma algebra containing M. Concretely, σ(M) is obtained by intersecting all sigma algebras that contain M. This “generated sigma algebra” is often described as the sigma algebra generated by M.

An example with a finite set {a, b, c, d} illustrates how σ(M) is forced to grow. If M consists only of the singletons {a} and {b}, then any sigma algebra containing M must also contain their complements, so {b, c, d} and {a, c, d} must appear. Closure under countable unions (which, in a finite setting, reduces to closure under finite unions) forces the union {a} ∪ {b} = {a, b} to be included. At that point, complements again force {c, d} to be included. Checking the sigma algebra axioms confirms the resulting collection is stable under complements and unions, and by construction it is the smallest possible sigma algebra containing the original singletons.

For infinite sets, the same idea works but the “generation” process can require infinitely many closure steps, making the resulting sigma algebra harder to describe explicitly. That sets up the most important sigma algebra in many analysis and probability settings: the Borel sigma algebra. When X is a topological space (or a metric space like R or R^N), the open sets are the basic objects of interest. The Borel sigma algebra, denoted B(X), is defined as the sigma algebra generated by all open sets. In practice, this means B(X) is the smallest sigma algebra that contains every open set, so it encodes the topological structure of X.

The Borel sigma algebra is large enough to include the sets typically needed for measurement, yet it is not as large as the full power set. That distinction matters because later, when defining measures, the required rules cannot be satisfied on arbitrary subsets of X. The Borel sigma algebra provides the right balance: it is sufficiently rich for real-world measurable sets while still being compatible with the measure axioms that come next.