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1. • CommentRowNumber1.
   • CommentAuthorDean
   • CommentTimeJun 30th 2025
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   Author: Dean
   Format: MarkdownItexI recently fixed my error in writing a proof of the existence iof Haar integral due to Uri Bader. It was interesting to me since it used the Krein-Milan theorem and Krein-Smulian theorem, so that the usual proof using radon measures and measure theory, and which relates to partitions of unity, isn't used. Compact Hausdorff groups feature in the Peter-Weyl theorem, and a corollary of that is that all such groups are pro-objects in the category of compact Lie-groups. The Peter-Weyl theorem uses the existence of Haar integral for compact Hausdorff topological groups, making the article on Haar integral an important one. Would it be possible to form stubs on the Krein-Milman and Krein-Smulian theorems, and, would it be possible for any of the members here to take a close look at my writeups of these two theorems?

   I recently fixed my error in writing a proof of the existence iof Haar integral due to Uri Bader. It was interesting to me since it used the Krein-Milan theorem and Krein-Smulian theorem, so that the usual proof using radon measures and measure theory, and which relates to partitions of unity, isn’t used.

   Compact Hausdorff groups feature in the Peter-Weyl theorem, and a corollary of that is that all such groups are pro-objects in the category of compact Lie-groups. The Peter-Weyl theorem uses the existence of Haar integral for compact Hausdorff topological groups, making the article on Haar integral an important one.

   Would it be possible to form stubs on the Krein-Milman and Krein-Smulian theorems, and, would it be possible for any of the members here to take a close look at my writeups of these two theorems?

2. • CommentRowNumber2.
   • CommentAuthorDean
   • CommentTimeJun 30th 2025
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   Author: Dean
   Format: MarkdownItexIt could also make sense to put them under an existing article.

   It could also make sense to put them under an existing article.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJul 1st 2025
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   Author: DavidRoberts
   Format: MarkdownItexNothing's stopping you writing the stubs here on the nForum and seeing who bites. I appreciate the fact that you checked first, which is more than some people do.

   Nothing’s stopping you writing the stubs here on the nForum and seeing who bites. I appreciate the fact that you checked first, which is more than some people do.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 1st 2025
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   Author: Todd_Trimble
   Format: MarkdownItexIn my opinion, it would be good to have material on Krein-Milman and Krein-Smulian, as these are notable results. I'd be willing to give feedback. One way to get started on Krein-Milman might be to create a section in [[locally convex topological vector space]], something like "Notable results", and then put [[Hahn-Banach theorem]] and [[Krein-Milman theorem]] as two of the bullet points. Then click on the grayed link for the second to create a new article. You can create a stub first, just stating it accurately to get started, and then shop around for references that have good accounts. If you felt moved to write up some exposition, that could be good as well. One thing to mention about Krein-Milman is its connection to the axiom of choice. Apparently the conjunction of that with the [[ultrafilter theorem]] is equivalent to AC. Meanwhile, existence and uniqueness of Haar measure do not require AC at all, and there are some good accounts of this, like [here](https://www.mscand.dk/article/view/10675/8696). I don't really know the lore, but the way you write, you don't seem keen on Radon measures and partitions of unity, and it makes me wonder why.

   In my opinion, it would be good to have material on Krein-Milman and Krein-Smulian, as these are notable results. I’d be willing to give feedback.

   One way to get started on Krein-Milman might be to create a section in locally convex topological vector space, something like “Notable results”, and then put Hahn-Banach theorem and Krein-Milman theorem as two of the bullet points. Then click on the grayed link for the second to create a new article. You can create a stub first, just stating it accurately to get started, and then shop around for references that have good accounts. If you felt moved to write up some exposition, that could be good as well.

   One thing to mention about Krein-Milman is its connection to the axiom of choice. Apparently the conjunction of that with the ultrafilter theorem is equivalent to AC. Meanwhile, existence and uniqueness of Haar measure do not require AC at all, and there are some good accounts of this, like here. I don’t really know the lore, but the way you write, you don’t seem keen on Radon measures and partitions of unity, and it makes me wonder why.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJul 1st 2025
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   Author: Urs
   Format: MarkdownItexcreated a stub entry *[[Krein-Milman theorem]]*

   created a stub entry Krein-Milman theorem

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 1st 2025
   • (edited Jul 1st 2025)
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   Author: Urs
   Format: MarkdownItexcreated a stub entry *[[Krein-Smulian theorem]]*

   created a stub entry Krein-Smulian theorem

7. • CommentRowNumber7.
   • CommentAuthorDean
   • CommentTimeJul 1st 2025
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   Author: Dean
   Format: MarkdownItexThanks for including the construction which doesn't use AC. I think I must have found the Krein-Milman theorem and Krein-Smulian theorem more intuitive, but it also assumes more. In broader terms I took an interest in "Hilbert's fifth problem from the other end", like T. Tao talks about here: https://terrytao.wordpress.com/2011/09/27/254a-notes-3-haar-measure-and-the-peter-weyl-theorem/ focusing on the construction of Haar integral for compact Hausdorff groups and the subsequent uniqueness theorem provided by the Peter-Weyl theorem that they are pro-compact-Lie groups as a fixed point of depth, and reflecting on ways to get there. Because of that theorem one can think backwards from Haar integral for pro-objects in the category of (smooth!) compact Lie-groups, specifically closed subobjects of certain pro-limits of unitary groups. Assuming these results, one can consider "Haar groups" (groups for which a unique Haar integral exists) and get that compact hausdorff groups are Haar groups using three further constructions: (1) Haar integral for $S^1$ (2) Haar integral for smooth Lie-groups using (1) (3) Haar integral for certain pro-constructions involving Lie-groups In the question I put forward on mathoverflow, I wanted to know about a kind of Ext¹-idea which would generalize the existence theorem for Haar integral for S¹. The way I do this it uses symmetrization $\frac{1}{n} \Sigma_{\zeta_n} f^{\zeta_n}$ and point-wise Cauchy sequences. That much revealed the more difficult Krein-Smulian theorem which uses net-closure of images in the weak topology and Tychonoff's theorem, as well as the equivalence of two kinds of closure for closures of convex subsets. If a Haar-group is a localic group for which there exists a unique Haar integral, then this shows that locally compact groups are Haar groups. But they are not the only ones: another source is Laurent series in two or more variables, which are not ever locally compact. This gets remedied in terms of the theory developed by John Tate (and clarified in Tom Leinster's post "Where do linearly compact vector spaces come from?"). I don't know whether those can be shown to be Haar groups (ones which are also Haar rings) without AC, or whether all Haar groups are such without assuming AC. I'll continue with this and also fix a further error I found that the convex hull needs to be closed in the current proof of the existence of Haar integral for a compactum on the Haar integral page.

   Thanks for including the construction which doesn’t use AC. I think I must have found the Krein-Milman theorem and Krein-Smulian theorem more intuitive, but it also assumes more.

   In broader terms I took an interest in “Hilbert’s fifth problem from the other end”, like T. Tao talks about here:

   https://terrytao.wordpress.com/2011/09/27/254a-notes-3-haar-measure-and-the-peter-weyl-theorem/

   focusing on the construction of Haar integral for compact Hausdorff groups and the subsequent uniqueness theorem provided by the Peter-Weyl theorem that they are pro-compact-Lie groups as a fixed point of depth, and reflecting on ways to get there. Because of that theorem one can think backwards from Haar integral for pro-objects in the category of (smooth!) compact Lie-groups, specifically closed subobjects of certain pro-limits of unitary groups.

   Assuming these results, one can consider “Haar groups” (groups for which a unique Haar integral exists) and get that compact hausdorff groups are Haar groups using three further constructions:

   (1) Haar integral for $S^1$

   (2) Haar integral for smooth Lie-groups using (1)

   (3) Haar integral for certain pro-constructions involving Lie-groups

   In the question I put forward on mathoverflow, I wanted to know about a kind of Ext¹-idea which would generalize the existence theorem for Haar integral for S¹. The way I do this it uses symmetrization $\frac{1}{n} \Sigma_{\zeta_n} f^{\zeta_n}$ and point-wise Cauchy sequences. That much revealed the more difficult Krein-Smulian theorem which uses net-closure of images in the weak topology and Tychonoff’s theorem, as well as the equivalence of two kinds of closure for closures of convex subsets.

   If a Haar-group is a localic group for which there exists a unique Haar integral, then this shows that locally compact groups are Haar groups. But they are not the only ones: another source is Laurent series in two or more variables, which are not ever locally compact. This gets remedied in terms of the theory developed by John Tate (and clarified in Tom Leinster’s post “Where do linearly compact vector spaces come from?”). I don’t know whether those can be shown to be Haar groups (ones which are also Haar rings) without AC, or whether all Haar groups are such without assuming AC.

   I’ll continue with this and also fix a further error I found that the convex hull needs to be closed in the current proof of the existence of Haar integral for a compactum on the Haar integral page.

8. • CommentRowNumber8.
   • CommentAuthorDean
   • CommentTimeJul 3rd 2025
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   Author: Dean
   Format: MarkdownItexI read today parts of the constructive paper in (4). I like that it seems to be about properties of a net induced by the group ring $\mathbb{C}[G]$ deepening the averaging $\text{sym}_{\zeta_n}$ in the case of $S^1$.

   I read today parts of the constructive paper in (4). I like that it seems to be about properties of a net induced by the group ring $\mathbb{C}[G]$ deepening the averaging $\text{sym}_{\zeta_n}$ in the case of $S^1$.

9. • CommentRowNumber9.
   • CommentAuthorDean
   • CommentTimeJul 5th 2025
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   Author: Dean
   Format: MarkdownItexSomething else I realized today is the following: in Tao's 2014 book, some of the material of which is discussed here: https://terrytao.wordpress.com/tag/locally-compact-groups/ He combines two important theorems of Gleason and Yamabe to obtain the result that, for $G$ any locally compact topological group, $G$ acts transitively on a discrete set $X$ with a stabilizer isomorphic to a pro-Lie group. Since $X$ and Pro-Lie groups both have elementary constructions as to their Haar integral, this opened up a nice possibility: to prove that all groups acting transitively on a discrete space with a stabilizer isomorphic to a pro-Lie group are Haar groups.

   Something else I realized today is the following: in Tao’s 2014 book, some of the material of which is discussed here:

   https://terrytao.wordpress.com/tag/locally-compact-groups/

   He combines two important theorems of Gleason and Yamabe to obtain the result that, for $G$ any locally compact topological group, $G$ acts transitively on a discrete set $X$ with a stabilizer isomorphic to a pro-Lie group.

   Since $X$ and Pro-Lie groups both have elementary constructions as to their Haar integral, this opened up a nice possibility: to prove that all groups acting transitively on a discrete space with a stabilizer isomorphic to a pro-Lie group are Haar groups.