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I am interested in examples of infinite-dimensional vector spaces that

  • are Banach spaces or even Hilbert spaces
  • are measure spaces

Instead of the full vector space, subsets with measure structure are interesting as well.

It is well-known that any non-trivial measure in the infinite-dimensional case cannot be as nice as the finite-dimensional Lebesgue measure. But perhaps there is an interesting measure that is not translation-invariant or defined on a mere subset.

What are examples for such spaces, showing up in research in particular?

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    $\begingroup$ Look up Wiener measure. $\endgroup$
    – KConrad
    Commented Nov 22, 2022 at 0:16
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    $\begingroup$ For every random variable $X$ with values in a space $E$, the distribution of $X$ is a measure on $E$. In particular, for any random process we get a probability measure on the space of its paths --- a space of sequences for a discrete-time process, and a space of functions of real variable for a continuous-time process. Are those measures interesting? That depends on whether you consider those random variables and processes interesting. $\endgroup$
    – user95282
    Commented Nov 23, 2022 at 11:02

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