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Apr
26
comment Applications of topology to discrete dynamical systems?
Possibly a google search for Ethan Akin "discrete" dynamical systems will lead to something.
Mar
10
comment does there exist a generalization of a manifold
See the nearly identically worded mathematics stackexchange question does there exist a generalization of manifold, which was asked 15 hours before this math overflow question was asked.
Feb
23
comment Predicates of infinite arity
Of possible interest: Relations of type $\alpha$ by Josef Šlapal (1988). Here the notion of an "$n$-fold relation" is generalized to an "$\alpha$-fold relation" for an ordinal number ${\alpha}.$ Most of the results relating to relations of type $\alpha$ (e.g. various types of inverses and compositions, and how they behave with respect to set operations such as union, intersection, set difference) involve various conditions on the ordinal $\alpha$ and conditions on one or more auxiliary ordinals.
Feb
17
answered A search for theorems which appear to have very few, if any hypotheses
Feb
10
awarded  Necromancer
Feb
9
comment A generalized ellipse
Interesting historical footnote: James Clerk Maxwell, On the description of oval curves, and those having a plurality of foci; with remarks by Professor Forbes, Proceedings of the Royal Society of Edinburgh 2 #28 (1845-466), 89-91. Forbes actually read the paper before the Royal Society of Edinburgh (on 6 April 1846) because Maxwell was not allowed due to his age. Maxwell was only 14 years old at the time.
Dec
7
comment Jarník-Besicovitch and outer measure
A possibly useful survey paper is Hausdorff dimension and Diophantine approximation by Maurice Dodson and Simon Kristensen (2003).
Dec
2
comment Jarník-Besicovitch and outer measure
I believe Jarnik proved much more precise results, so you may want to dig up his papers or look for commentaries on what he did. I posted one such "more precise" result in this 25 March 2001 sci.math post (scroll down to where "Jarnik's theorem (simplified):" appears on the left side).
Nov
17
comment A metric on the set of BV functions, is it mentioned/studied in literature?
I believe what you're interested in, but for $||x-y||_{L^1}$ instead of $||x-y||_{L^2},$ can be found near the bottom of p. 422 of The space of functions of bounded variation and certain general spaces by Clarence Raymond Adams [Trans. AMS 40 (1936), 421-438]. See also Abstract #1 on p. 19 here (1937) and Abstract #1 on p. 27 here (1940).
Oct
19
comment Can we find minimal-diameter metrics without computability?
Although I don't know anything about this topic, that didn't stop me from posting a comment about it to the math StackExchange question Connections of theory of computability and Turing machines to other areas of mathematics. Maybe Soare's 3 August 1999 FOM post that I cite, which of course does not answer your question, will provide additional insight into the topic.
Sep
25
comment Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Possibly of interest: [A] Utpal Kumar Bandyopadhyay, On vector measures with the Darboux property, Quarterly Journal of Mathematics (Oxford) (2) 25 (1974), 57-60; [B] Hwang-Wen Pu and Huay-Min Huoh, Darboux property for transformations, Journal of Mathematical Analysis and Applications 90 #2 (December 1982), 299-306.
Sep
21
comment Relative null-ness
You might want to look at Borel's "rarefaction" classification of measure zero sets. See Winfried Just and Claude Laflamme's 1990 Trans. AMS paper Classifying sets of measure zero with respect to their open covers. Laflamme wrote at least two more papers related to this, and you can find earlier work by Frechet, Zenon Moszner, Léonard Urbanek, Claude Tricot, Frédéric Roger, and some others. (I have a bibliography on this topic if you're interested.)
Sep
18
revised About the generating structure of Borel field
I have given a more complete proof of "Prop 3", which was asked for in the comments section.
Sep
18
suggested approved edit on About the generating structure of Borel field
Sep
16
comment About the generating structure of Borel field
For some references about these issues, see Is there any good text introducing a part of Borel-hierarchy which is in need in measure theory.
Sep
16
comment About the generating structure of Borel field
In most "natural situations", each level is a proper subset of the next level (Borel sets in a complete metric space with no isolated points, for example), but of course in sufficiently artificial examples (e.g. finite $\sigma$-algebras) this might not be the case. The proofs I've seen that each level is strictly contained in the next level tend to be a bit technical, however. On the other hand, if by "full" you mean all the sets in the $\sigma$-algebra occur after $\omega_1$ levels, this is fairly straightforward to show using your "Hint".
Sep
15
comment Mathematical writing : using an “out-of-date” notation
A google-books search using the word "function" and the phrase "of class $k$" will show you "class $k$" is used in a variety of settings. Notation conventions tend to come and go, but I'm willing to bet that "of class $k$" will be a lot less meaningful 50 years from now than "of class $C^{k}$".
Sep
15
comment About the generating structure of Borel field
Regarding Prop 3, I believe you have the inclusions pointing in the wrong direction in the first two lines of your proof. Also, you'll probably need to use transfinite induction to show the first inclusion. Specifically, show that for each countable ordinal $\beta,$ we have $A_{\beta} \subset \sigma(A),$ and once this is done the inclusion involving the union follows. Also, for the second inclusion you'll need to show that the union is actually a $\sigma$-algebra (the hint is helpful here) in order to show it contains $\sigma(A).$
Aug
19
comment What results exist for functions with regionally fluctuant fractal dimension?
Something else possibly relveant: Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure by Guy David and Stephen Semmes (1997). See the review by Kenneth John Falconer and the Zbl 887.54001 review.
Aug
18
comment What results exist for functions with regionally fluctuant fractal dimension?
Possibly relevant to what you're looking for: How can dimension depend on the point?