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Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some concrete examples of:

  • Locally-positive Borel probability measures on $(\prod_{n \in \mathbb{N}} \mathbb{R},B)$,
  • Locally-positive $\sigma$-finite (but not finite) Borel measures on$(\prod_{n \in \mathbb{N}} \mathbb{R},B)$?

How does the situation change when the product is indexed over $\mathbb{R}$?

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  • $\begingroup$ There is also this construction: eudml.org/doc/281217 $\endgroup$
    – ABIM
    Feb 10, 2020 at 21:20
  • $\begingroup$ The situation changes dramatically when the product is indexed over $\mathbb{R}$, as the product space is no longer Polish. There are tons of interesting probability measures on the countable product of $\mathbb{R}$ (indeed, the study of these is practically the entire field of probability theory), and basically none on the uncountable product. $\endgroup$
    – Nate Eldredge
    Feb 10, 2020 at 22:30
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    $\begingroup$ Wait a second - your bullet points have a product over $n \in \mathbb{R}$. Is that a typo? Are you mainly interested in countable or uncountable products, or both? $\endgroup$
    – Nate Eldredge
    Feb 10, 2020 at 22:37
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    $\begingroup$ Here's a construction working for both cases (countable and continuum product): the product is then separable (though non-metrizable for an uncountable product), so there a dense sequence $(x_n)$, then the measures $\sum\delta_{x_n}$ and $\sum 2^{-n}\delta_{x_n}$ are positive on nonempty open subsets, are $\sigma$-finite and finite respectively. (Of course they have atoms, but this is not excluded.) $\endgroup$
    – YCor
    Feb 11, 2020 at 1:28
  • $\begingroup$ @NateEldredge This was a typo, I was initially interested in products over $\mathbb{R}$ both $\mathbb{N}$ but had simplified the question rapidly after to $\mathbb{N}$. $\endgroup$
    – ABIM
    Feb 11, 2020 at 8:39

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$\newcommand\R{\mathbb R}$ Let $T$ be any countable nonempty set. Let $B$ be the Borel $\sigma$-algebra over $\R^T$ generated by the Tikhonov product topology on $\R^T$. Take any $t_0\in T$. For each natural $k$ and each $t\in T$, let $$\nu_{k,t}:= \begin{cases} N(0,1)&\text{ if } t\ne t_0,\\ N(k,1)&\text{ if } t=t_0. \end{cases} $$

By Kolmogorov's measure extension theorem , for each natural $k$ there is a product probability measure $$\mu_k:=\bigotimes_{t\in T}\nu_{k,t} $$ on $B$, which will obviously be locally positive.

Moreover,
$$\mu:=\sum_{k=1}^\infty\mu_k$$ will be a locally-positive $\sigma$-finite (but not finite) measure on $B$.

The same constructions will work for uncountable sets $T$ if, instead of the Borel $\sigma$-algebra $B$, we will take the $\sigma$-algebra generated by the standard base of the Tikhonov product topology.

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    $\begingroup$ Indeed, the Kolmogorov extension theorem is the main tool for producing probability measures on this space, and is probably the first thing the OP ought to study if not already familiar with it. $\endgroup$
    – Nate Eldredge
    Feb 10, 2020 at 22:29
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    $\begingroup$ But AFAIK, if $T$ is uncountable then Kolmogorov does not give an extension to the Borel $\sigma$-algebra, only to the product $\sigma$-algebra, which is much smaller. I do not know of any way to produce nontrivial measures on the Borel $\sigma$-algebra of an uncountable product of $\mathbb{R}$, except using things like measurable cardinals. $\endgroup$
    – Nate Eldredge
    Feb 10, 2020 at 22:33
  • $\begingroup$ @NateEldredge : You are right. I have now corrected this. $\endgroup$
    – Iosif Pinelis
    Feb 10, 2020 at 22:49
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    $\begingroup$ @YCor: I really shouldn't have said that because I don't actually know what I'm talking about. All I mean is that in general I don't know of "interesting" ways to produce measures on such large spaces, certainly not any that give us such a rich class of measures as for countable products, and that the only examples I know of "large" spaces with really nontrivial measures are things like measurable cardinals. $\endgroup$
    – Nate Eldredge
    Feb 11, 2020 at 13:51
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    $\begingroup$ @MrMMS: Usually a good resource is cantorsattic.info, but it seems to be down at present. Otherwise there's Wikipedia. But as mentioned I am really not a good person to ask about such things. $\endgroup$
    – Nate Eldredge
    Feb 11, 2020 at 13:54

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