Describe information with sigma-algebras

Try

A sensor reports only whether a number on a six-outcome device is at most two. From this observation one can distinguish the event A consisting of the first two outcomes and its complement, but not the individual outcomes inside either group. The available yes-or-no questions are empty, A, its complement and the whole space. A sigma-algebra formalizes a collection of events whose logical combinations remain meaningful.

Predict

Predict what happens when a second sensor reports whether the outcome is even. Intersections of the two sensor events and their complements produce smaller distinguishable groups called atoms. Every event determined by the observations is a union of these groups. Counting atoms rather than original outcomes gives the size of the generated finite sigma-algebra. Empty intersections are omitted; an impossible observation combination is not a new atom.

Experiment

For outcomes 1 through 6, compare the two observations.

First sensor	Even sensor	Atom
At most 2	Even	2
At most 2	Odd	1
Above 2	Even	4,6
Above 2	Odd	3,5

There are four nonempty atoms and therefore sixteen distinguishable events. Construct an event that separates outcome two from four, then explain why no combination of these observations separates four from six.

Understand

A sigma-algebra F on a set X contains X, is closed under complements relative to X, and is closed under countable unions. It consequently contains the empty set and is closed under countable intersections by De Morgan’s law. For a family C of subsets, its generated sigma-algebra σ(C) is the intersection of all sigma-algebras containing C; the power set guarantees that this collection is nonempty. Intersections preserve the axioms, proving that σ(C) exists and is the smallest such sigma-algebra. This construction defines minimality without claiming there is a finite algorithm that lists every generated set.

Apply

On a topological space, the Borel sigma-algebra is generated by the open sets. On the real line, open intervals with rational endpoints suffice because every open set is their countable union. Countability is crucial: a sigma-algebra need not contain arbitrary unions of its sets. For example, the countable-cocountable sigma-algebra on the real line contains every singleton, but not an interval with both it and its complement uncountable. The interval is an uncountable union of singletons. Thus a measurable-event structure need not be a topology, and a topology need not itself be closed under complements.

Continue

A measurable space is only the pair (X, F); no sizes have yet been assigned. A measure will add numerical values while respecting countable disjoint unions. Keep this order of definitions clear: deciding which sets are measurable and deciding how much they measure are separate operations. The real-analysis sequence supplies countability and limit prerequisites. This course uses the undergraduate scope of Warwick ST 350, 2026, with the MIT 18.125 Spring 2010 schedule as a separately labeled advanced comparison, not a claim of institutional equivalence.

Continue to Derive continuity from countable additivity.