Why countable additivity

A well-defined probability measure must have the property that where is a sequence of mutually exclusive events i. In other words, the probability of a countable union of disjoint events must be equal to the sum of their probabilities. It is easy to prove that countable additivity implies finite additivity, that is, for any set of mutually exclusive events. Note that for any.

Therefore, we can set for in the definition of countable additivity. It cannot be proved. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

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Terence Tao's free book on measure theory spends some time near the beginning developing "Jordan measure", which is a sort of finitely-additive version of Lebesgue measure. As he points out, that theory is mostly fine as long as one happens to only work with things that are Jordan measurable.

However, as Tao proves in Remark 1. In general, I think Tao's presentation does show clearly the similarites and differences between Lebesgue and Jordan measure, although it takes some mathematical maturity to read it, so it might not help your friends. The key benefit of countable additivity is that once open intervals are measurable, all Borel sets are measurable and, moreover, all analytic sets - continuous images of Borel sets - are Lebesgue measurable.

So, unless we really try, we are unlikely to construct nonmeasurable sets. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Ask Question. Asked 8 years ago. Active 7 years, 11 months ago. Viewed 2k times. Jakub Konieczny Jakub Konieczny 12k 1 1 gold badge 28 28 silver badges 68 68 bronze badges.

Typical ones fail countable additivity; the exceptions are either trivial or else live on large cardinals. Add a comment. Active Oldest Votes. Here are the de Finetti references: de Finetti, Bruno, Carl Mummert Carl Mummert Why must probability be geometrical? I can only claim to address two parts. Tao's book presents one in detail. In pure probability, ignoring integration, we want to look at many Borel sets that are not Jordan measurable. That may be an argument for countably additive probability in cases where there is a natural geometrical interpretation of probability space.

But on the face of it probabilities can make sense for arbitrary events, not having any connection to the space and the classical measure problem. If you can connect probability to the measure problem then I think you will actually have an answer to most of the question.

The idea is that if a point is in an open set, we will eventually realize this by computing sufficiently close approximations of the point. However, that still ties things to Euclidean space in a sense.