Probability spaces and random elements

The outcome space contains elementary possibilities; its elements are not themselves automatically events. Events are selected subsets in a sigma-algebra, which includes the whole space and is closed under complements and countable unions. These closures allow a sequence of yes-or-no questions to form a new legitimate question. Countability matters because limits of observations often use infinitely many operations. No requirement of closure under every uncountable union is imposed, and on uncountable spaces that stronger requirement would change the available measure theory.

A probability measure is nonnegative, assigns total mass one, and adds the masses of countably many pairwise disjoint events. From these assumptions follow monotonicity, the complement rule, and continuity under increasing unions. The last property explains why finite approximations can determine a limiting event. Specifying just finite additivity would not provide the same limit passage. In a finite experiment every subset can be measurable, but this convenient special case should not be mistaken for the definition of probability on general spaces.

A random element is a measurable map from the outcome space to another measurable space. Its values might be real numbers, vectors, functions, or labels. Measurability means that every permitted question about its output pulls back to an event on the original space. It does not mean that the map is continuous, numerically smooth, or unpredictable. A deterministic constant is a valid random variable, and a discontinuous indicator is measurable whenever its defining event is measurable.

For a real-valued map, it suffices to verify that inverse images of half-lines are events, because these half-lines generate the Borel sigma-algebra and inverse images preserve the relevant set operations. A map that splits an atom of a coarse sigma-algebra cannot be measurable into the usual real Borel space. Here an atom means a nonempty measurable cell with no nontrivial measurable subset within it. The obstruction is informational: the output would reveal a distinction that the original model does not permit us to ask about.

Consider the uniform probability measure on the unit interval. Every particular point has probability zero, yet an outcome is still some point. Probability zero therefore means negligible for the measure, rather than logically impossible. Likewise an almost-sure assertion may fail on a measurable null set. This distinction is essential when comparing random variables: two variables equal almost surely can differ as functions at particular outcomes without changing integrals of their integrable versions.

A probability space is complete when every subset of a measurable null set is measurable. Completing a space adds such subsets without changing probabilities of previously measurable events. Completeness is useful for describing modifications on negligible sets, but is an additional property rather than one of the three probability axioms. In particular the Borel sigma-algebra equipped with a measure need not already contain every subset of every null set. Always identify whether the discussion uses the original sigma-algebra or its completion.