Sigma algebras and measurable maps

A sigma algebra on X contains X and is closed under complements and countable unions. These rules also give the empty set, countable intersections, and set differences. Countability is essential: a countable sequence of observations must still define an observation, but arbitrary unions need not. Intersecting every sigma algebra containing a family C produces the generated sigma algebra. This construction proves existence even when listing all its sets is impractical. It is the smallest collection satisfying the closure rules and containing the original questions.

On a finite space, a partition provides a concrete picture. If the observer can distinguish the blocks {a,b} and {c,d} but cannot distinguish points within either block, the measurable sets are the empty set, those two blocks, and the whole space. Adding the question {a} refines the information: complements and intersections now separate a from b. No numerical weights were needed. A measure is additional data assigning nonnegative values to these sets, with countable additivity on disjoint unions.

For measurable spaces X and Y, a function f is measurable when the inverse image of every measurable subset of Y is measurable in X. The sigma algebra on each side is part of the assertion. Inverse images commute with complements and arbitrary unions, explaining why this definition works so cleanly. Direct images do not generally preserve complements or intersections. Consequently a measurable map need not carry every measurable input set to a measurable output set. Confusing these directions obscures the actual theorem.

For real outputs equipped with their Borel sigma algebra, it suffices to test the sets where f exceeds a rational threshold. Such half-lines generate all Borel sets. More generally, if G generates the target sigma algebra, collect all target sets whose inverse images are measurable. That collection is itself a sigma algebra, so containing G forces it to contain the generated sigma algebra. This short argument replaces an impossible test over every Borel set with a manageable family of inequalities.

Compositions of measurable maps are measurable because the inverse image of a composition is an iterated inverse image. For real measurable functions, sums, products, and pointwise limits are measurable as well. For limits, the event that a limiting value exceeds a threshold can be written using countable unions and intersections of threshold events for the sequence. These closure results justify building complicated observables from elementary ones. They assert that the result is eligible for integration, not that its integral is finite.

After choosing a measure, completeness asks a further question: is every subset of every measurable null set also measurable? Completing a measure space adds those subsets and gives them measure zero. This enlargement need not add any new integral values for previously measurable integrable functions. However, it changes which representatives are measurable after modifications on null sets. Borel measurability and Lebesgue measurability on the real line therefore should not be silently interchanged: Lebesgue measure includes the completion of the Borel measure.