Before we can look at how Kolmogorov reformulated probability (the Kolmogorov axioms<\/em>), we need to look at just what a measure space is.<\/p>\n A measure space is just a set with a measure function<\/em>. So let X<\/b> be a set. A measure<\/em> μ on X<\/b> is a function from a subset of X<\/b> to a real number: So the idea is pretty simple: a measure space is just a way of defining the size<\/em> of a subset in a consistent way.<\/p>\n To work with probability, you need a measure space where the measure of the entire set is 1. With that idea in mind, we can put together a proper, formal definition of a probability space that will really allow us to work with, and to combine probabilities in a rigorous way.<\/p>\n Like our original version, a probability space has a set of events, called its event space<\/em>. We’ll use F<\/b> to represent the set of all possible events, and e<\/em> to represent an event in that set.<\/p>\n There are three fundamental axioms of probability, which are going to look really<\/em> similar to the three axioms of a measure space:<\/p>\n This is very similar to the informal version we used earlier. But as we’ll see later, this simple formulation from measure theory will give us a lot of additional power.<\/p>\n It’s worth taking a moment to point out two implications of these axioms. (In fact, I’ve seen some presentations that treat some of these as additional axioms, but they’re provable from the first three.<\/p>\n The brilliance of Kolmogorov was realizing that these rules were everything you need<\/em> to work out any probability you want – in both finite and infinite spaces. We’ll see that there’s a lot of complexity in the combinatorics of probability, but it will all always ultimately come back to these three rules.<\/p>\n","protected":false},"excerpt":{"rendered":" The way that I’ve talked about probability so far is mostly informal. That’s the way that probability theory was treated for a long time. You defined probability spaces over collections of equal probability sets. You combined probability spaces by combining their events into other kinds of equally probable events. The problem with that should be […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[53],"tags":[],"class_list":["post-2219","post","type-post","status-publish","format-standard","hentry","category-probability"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-zN","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/2219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=2219"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/2219\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=2219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=2219"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=2219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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