I'm learning about infinite dimensional probability, and most resources I've consulted so far motivate things by saying there is no infinite dimensional Lebesgue/translation invariant measure that gives nontrivial measures for unit balls in a Banach space. The proposed alternative is almost always to look Gaussian measures instead. Why are they seen as the standard alternative to Lebesgue measure when doing probability in infinite dimensions?
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1Because they are invariant with respect to unitary operations (as is easy to prove and well known). – Salcio Feb 06 '24 at 23:13