a measurable space

Before you can talk about how likely things are, you first have to settle two prior questions: what are the possible outcomes, and which collections of outcomes are we allowed to ask about. A measurable space packages exactly those two pieces of bookkeeping and nothing more. It is the stage and the cast of allowable questions, set up before any probabilities are assigned. Crucially, it carries no notion of likelihood yet — it only says which events exist as legitimate things to measure.

Formally, a measurable space is a pair (Omega, F): a set Omega of all possible outcomes, together with a sigma-algebra F of subsets of Omega. The members of F are called the measurable sets, or in a probability context the events. So the space tells you the universe of outcomes and the agreed-upon family of events, with the family guaranteed to be closed under complement and countable union. Think of it as a blank measuring apparatus: the dial positions (the events) are marked out, but no readings (probabilities) have been taken. A function between two measurable spaces is called measurable if it respects this structure — the preimage of every event in the target is an event in the source.

Separating this skeleton from the actual measure is one of the quiet but powerful moves of the rigorous theory. The same measurable space (R, Borel sets) can host countably many different probability measures — the normal, the exponential, the uniform on [0,1] — each one a different way of distributing weight over the same set of events. Pinning down outcomes and events first, then layering probability on top, is what lets the subject handle infinite and continuous models without contradiction.