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I'm in over my head here, but I am wondering about the probability that a distribution has finite variance? (or a finite mean?)

By this, I don't mean that there is some set of data, just over the set of all possible continuous distributions, is the subset of distributions with finite variance of measure 1, or 0? (And the same regarding finite mean.)

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You need to define a probability distribution over probability distributions for this question to make sense. – Qiaochu Yuan May 7 '13 at 3:21
OK, what measures can be applied to the set of distributions? Aren't they functions in R^2 with a restriction of summing to 1? – David Manheim May 21 '13 at 15:39

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