Talk:Atom (measure theory)

Blanked sections

There seeems to be something wrong in this article, right after "This measure has no atoms". — Preceding unsigned comment added by 157.92.4.72 (talk) 21:07, 13 October 2011 (UTC)

Thank you for your note. I have now restored two sections that had been removed by an anonymous user several weeks ago without anybody noticing. — Tobias Bergemann (talk) 07:53, 14 October 2011 (UTC)

Incorrect reference

According to the article of Sierpinski is only proving a restricted version of what is called Sierpinski's article in this wiki entry. Is this relevant? Leonry (talk) 17:07, 21 May 2019 (UTC)

I read that mathexchange question 23 and 1/2 times and it doesn't make sense to me. Just pick a set $E_{2}$ with measure zero, what's the problem? I mean, Sierpinski proves a result for $E_{0}$ a subset of Cartesian space $R^{n}$ and Cartesian space is full of points, which are necessarily of measure zero. I mean, you can assign a non-zero measure to at most a finite number of points (or dice it up into a finite number of atomic parts). So surely, I can just "pick one" that's measure zero!? So is the math-exchange question secretly about the axiom of choice, about the difficulty of finding some point, any point, with a measure of zero? Perhaps something about Zorn's lemma, and Sierpinski's use of it, or failure to use it? Was the questioner too shy to come out and say so? ZFC seems like a plausible foundation? Perhaps I am being a complete moron and am missing something painfully obvious? If there's actually a problem there, it is not posed clearly enough to be understandable. 67.198.37.16 (talk) 06:27, 13 September 2020 (UTC)

References

Atomic don't implies sum of delta functions

In [0,1] consider the sigma-algebra of countable and co-countable subsets. Countable has measure 0, co-countable measure 1. Then co-countables forms the only atomic class, the measure is atomic, but it is not a sum of delta functions — Preceding unsigned comment added by 83.32.52.102 (talk) 20:29, 6 December 2020 (UTC)

Linking to "Stack Exchange" basic question, "homework"

It doesn't seem right to link the Wiki article its definition to a "Stack Exchange" article question with little comment.

Also there's not an answer, ....

Clearly these atoms in measure have sigma algebras. 97.113.48.144 (talk) 04:44, 29 September 2022 (UTC)

Atom (order theory)

Any relation with atoms of $\sigma$-algebras (order-theoretic atoms)? Thatsme314 (talk) 11:31, 5 June 2023 (UTC)

Atoms of $\mu$ = minimal elements of $\{A\in {\cal {A}}:\mu (A)\neq 0\}$ (ordered by set inclusion).

Atoms of ${\cal {A}}$ = minimal elements of $\{A\in {\cal {A}}:A\neq \emptyset \}$ (ordered by set inclusion).

It seems that both notions of atom capture some notion of "minimal non-trivial measurable sets", where "non-trivial" means $\mu (A)\neq 0$ for "measure atom"s and $A\neq \emptyset$ for "$\sigma$-algebra atom"s.

The former notion generalizes the latter in a straightforward way: the atoms of ${\cal {A}}$ are precisely the atoms of the restriction/subspace measure $\mu |_{\cal {A}}$, where $\mu$ is counting measure on $X=\cup {\cal {A}}$. Thatsme314 (talk) 17:30, 5 June 2023 (UTC)

References