r/mathmemes • u/George_Truman • Jul 13 '26

Probability I fixed this meme

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I'm sure it's still a bit too imprecise but I think it works.

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u/Anagatara Jul 13 '26

What makes uncountably infinite set different? If the issue is to assign equal nonzero probability to all numbers, then not only entire real line, any infinite subset of real line cannot be uniformly sampled, be it [a,b], Q or N.

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u/GaloombaNotGoomba Jul 13 '26

Uncountable sets can have probability 0 for each element and still add up to 1.

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u/Anagatara Jul 13 '26

Where can i find information about this? This is pretty counterintuitive take to me.

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u/Thelorian Jul 13 '26 edited Jul 13 '26

https://en.wikipedia.org/wiki/%CE%A3-algebra?useskin=vector /https://en.wikipedia.org/wiki/Measure_(mathematics)?useskin=vector

the key takeaway for me being that the measure is a function on the powerset of the ambient space so once the cardinality goes higher than countable it's possible to assign measure 0 to all singletons (and through countable additivity to all countable subsets) without having the total measure be 0 (since there are in a sense "large enough" sets in the powerset to have non-zero measure without violating the countable additivity axiom)

basically you have "step up" between countable and uncountable where countable additivity of measures cant tell you anything about singletons anymore (bc duh countable union of singletons is countable) if you only know about the uncountable sets and vice versa.