Large cardinal axioms are very strong hypothesizes and as any other strong hypothesis they have many strange consequences in mathematics.

On the other hand we know that if we bring even the least infinity ($\aleph_0$) into the story, there will be many strange theorems in probability which are challenging to our intuition. (Recall the well-known example of a monkey who is typing randomly and can type entire masterworks of human literature if he has an infinite time).

I wonder what would happen if we work with large cardinalities rather than $\aleph_0$ in probability?

In fact there are some connections between large cardinals and probability. Probably the most well known is the problem of extending Lebesgue measure to all sets of reals which is equivalent to the existence of a real valued measurable cardinal $\leq 2^{\aleph0}$. Note that a real valued measurable cardinal $\kappa$ by definition is an uncountable cardinal in which there is a $\kappa$-additive probability measure on the power set of $\kappa$ which vanishes on singletons.

Question: What are examples of theorems in probability theory which are related to large cardinals or have a large cardinal strength? Is there any example of a philosophically strange probabilistic phenomena which happens in the real world if we assume existence of large cardinals? I mean something like the example of a monkey who can type all texts of human literature if we assume existence of $\aleph_0$ as the weakest large cardinal assumption.

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    $\begingroup$ journals.cambridge.org/action/… ${}\qquad{}$ $\endgroup$ – Michael Hardy Nov 2 '14 at 18:34
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    $\begingroup$ The word 'provability' in fourth line looks like a misspelling of 'probability'. Am I right? $\endgroup$ – Hanul Jeon Nov 3 '14 at 1:45
  • $\begingroup$ @tetori Edited. Thanks for your useful comment. $\endgroup$ – user180918 Nov 5 '14 at 22:40
  • $\begingroup$ A similar question on MathOverflow: Infinite cardinals and probability theory $\endgroup$ – Martin Sleziak Apr 25 at 4:43
  • $\begingroup$ Can an uncountable set have anything to do with probabilities? Is sounds like a contradiction. $\endgroup$ – Shimon S Aug 5 at 6:21

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