User felipeg - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:04:58Z http://mathoverflow.net/feeds/user/18384 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128259/is-function-from-topological-group-to-metric-space-borel Is function from topological group to metric space Borel? FelipeG 2013-04-21T16:52:51Z 2013-04-21T19:05:58Z <p>Let $G$ be a pseudometrizable compact abelian topological group, $X$ a compact metric space and $f:X\rightarrow G$ a continuous bijective function.</p> <p>Suppose there exists $g\in G$ such that if $d_{G}(g_{1},g_{2})\leq\epsilon$ then there exists $n$ such that $d_{X}(f^{-1}g^{n}g_{1},f^{-1}g^{n}g_{2})\leq\epsilon.$</p> <p>If $G$ is not metrizable then in general $f^{-1}$ is not continuous but can we conclude $f^{-1}$ is Borel? </p> http://mathoverflow.net/questions/128183/existence-of-limit-measure Existence of limit measure FelipeG 2013-04-20T17:46:28Z 2013-04-20T20:50:45Z <p>Let $X$ be a separable metric space, $\mu_{n}$ a sequence of Borel probability measures and $\mathcal{C}$ be a family of sets that is closed under finite unions and interections, and that contains all the balls. If $\mu_{n}(A)$ converges for every $A\in\mathcal{C}$, does there exists a Borel measure $\mu_{\infty}$ such that $\mu_{\infty}(E)=\lim\mu_{n}(E)$ for every $E\in\mathcal{C}?$</p> <p>From Theorem 4.3 in <a href="http://msp.org/pjm/1964/14-3/pjm-v14-n3-p23-p.pdf" rel="nofollow">this paper</a> we can get this result when $X$ is locally compact. Here Sion makes an outer measure and then shows open sets are measurable. His proof definitively uses local compactness. </p> <p>Does anybody know if the result is true when $X$ is not necessarily locally compact?</p> http://mathoverflow.net/questions/123252/extension-of-measures-from-the-ball-sigma-algebra-to-the-borel-sigma-algebra Extension of measures from the ball sigma-algebra to the borel sigma-algebra FelipeG 2013-02-28T18:32:52Z 2013-02-28T19:52:30Z <p>Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and $\Sigma_{2}$ the sigma algebra generated by balls (open and closed). </p> <p>If $\mu$ is a probability measure on $\Sigma_{2}$ can it be extended to a measure on $\Sigma_{1}?$</p> http://mathoverflow.net/questions/121036/do-ergodic-isometries-have-discrete-spectrum Do ergodic isometries have discrete spectrum? FelipeG 2013-02-07T03:19:11Z 2013-02-07T03:19:11Z <p>Let $X$ be a metric space, $\mu$ a Borel probability measure, and $T:X\rightarrow X$ be an ergodic measure preserving isometry. </p> <p>Is $(X,\mu,T)$ measure theoretically isomorphic to a minimal isometry on a compact metric space (equivalently it has discrete spectrum)?</p> <p>I understand Krieger representation theorem states that ergodic MPT are measure theoretically isomorphic to minimal systems on a compact metric spaces. I would like to know if structure like isometry can be conserved. </p> <p>The general question I am interested in is: do ergodic isometries have discrete spectrum?</p> http://mathoverflow.net/questions/111949/weak-convergence-and-cesaro-convergence-of-mu-n-e-imply-convergence-of-mu Weak convergence, and Cesaro convergence (of mu_n (E) ) imply convergence (of mu_n (E))? FelipeG 2012-11-09T23:16:57Z 2012-11-12T19:55:25Z <p>Let $X$ be a compact metric Borel space. Suppose $\mu_{n}(A)\rightarrow\mu(A)$ for all $\mu-$continuity sets $A$ (sets with zero boundary measure), where $\mu_{n}$ is a sequence of probability measures. (some people call it weak other weak* convergence)</p> <p>If $E$ is a measurable set such that $\mu(E)>0$ and the Cesaro average of $\mu_{n}(E)$ converges; can we conclude that $\mu_{n}(E)$ converges? </p> <p>Can we conclude this with extra hypothesis?</p> <p>I am particularly interested in the case when $T:X\rightarrow X$ is a continuous transformation, $\mu_{n}=T^{n}\mu_{1},$ and $E=\cap T^{-i}A_{i}$ where $A_{i}$ is a sequence of $\mu-$continuity sets. </p> http://mathoverflow.net/questions/48771/proofs-that-require-fundamentally-new-ways-of-thinking/94167#94167 Answer by FelipeG for Proofs that require fundamentally new ways of thinking FelipeG 2012-04-16T01:00:28Z 2012-04-16T01:00:28Z <p>The first formal proofs using limits. (the oldest ones I know are in Newton's Principia)</p> http://mathoverflow.net/questions/87952/a-system-of-equations-for-integers A system of equations for integers FelipeG 2012-02-09T02:20:25Z 2012-02-10T23:43:52Z <p>Working with cellular automata I came across a system of equations for unknown integers $R_{k}$ and $C_{k}$ that looks like this.</p> <p>$\binom{m}{k}=R_{k}+C_{k}+\sum\limits_{j=1}^{k-1}R_{j}C_{k-j}.$</p> <p>Where 0&lt; k$\leq$ 2m </p> <p>(for k>m we take $\binom{m}{k}=R_{k}=C_{k}=0$)</p> <p>Given $R_{1}$, the system has a unique solution. </p> <p>Has anyone seen something similar? Do you know if it still possible to solve it, if instead of $\binom{m}{k}$ in the left you put something else? </p> <p>I just want to know if it's related to something else, and if it's possible to solve more general systems. </p> http://mathoverflow.net/questions/79531/bijective-function-on-a-dense-set Bijective function on a dense set FelipeG 2011-10-30T18:28:23Z 2011-11-03T14:54:41Z <p>Suppose X is a complete metric space, and $f:X↦X$ a continuous surjective function. Let D be a dense set. Suppose $f:D↦D$ is injective and $f^{-1}(D)=D$. </p> <p>Is $f$ injective ?</p> <p>Is there a family of metric spaces where you can conclude $f$ is injective?</p> http://mathoverflow.net/questions/128259/is-function-from-topological-group-to-metric-space-borel/128277#128277 Comment by FelipeG FelipeG 2013-04-22T18:27:40Z 2013-04-22T18:27:40Z Thanks for your comment. There is something I don't understand. Are you assuming the closure of the identity is not a single point? Is this is necessarily true? http://mathoverflow.net/questions/128259/is-function-from-topological-group-to-metric-space-borel Comment by FelipeG FelipeG 2013-04-22T18:13:55Z 2013-04-22T18:13:55Z Sure. The multiplication of g is a dynamical system, a rotation in a compact abelian group. I want to construct a dynamical isomorphism from a dynamical system in X to G. http://mathoverflow.net/questions/128183/existence-of-limit-measure Comment by FelipeG FelipeG 2013-04-21T17:18:41Z 2013-04-21T17:18:41Z Thanks for your comments. Theorem 4.3 talks about two kinds of limits. The other limit is the outer measure constructed in section 3. Theorem 3.3 mentions that the outer measure is Radon (hence every borel set is measurable) if every open set is the countable union of compact sets. http://mathoverflow.net/questions/121036/do-ergodic-isometries-have-discrete-spectrum Comment by FelipeG FelipeG 2013-02-07T08:41:57Z 2013-02-07T08:41:57Z Sure, a theorem by Halmos and Von Neumann state that T is transitive (one dense orbit) and isometric, iff it is topologically isomoprhic to a minimal rotation on a compact abelian group iff T is minimal and has discrete topological spectrum. (topological because in this case the operator associated to the dynamical system is acting on the space of continuous functions. ) A good reference for this is An introduction to ergodic theory by Peter Walters. http://mathoverflow.net/questions/119133/a-similar-result-to-lusins-theorem-to-characterize-borel-measures Comment by FelipeG FelipeG 2013-01-17T18:10:45Z 2013-01-17T18:10:45Z Thank you Bill. Andreas: I am mostly curious about metric spaces but I don't think that assumption is necessary. http://mathoverflow.net/questions/111949/weak-convergence-and-cesaro-convergence-of-mu-n-e-imply-convergence-of-mu/111960#111960 Comment by FelipeG FelipeG 2012-11-12T20:05:31Z 2012-11-12T20:05:31Z Thanks, I was not looking for the case when the cesaro means do not converge. I wanted that as a part of the hypothesis. But I think your counter example works, if you take $x_{n}$=0 only sporadically, then you will have cesaro mean convergence but not convergence. http://mathoverflow.net/questions/101593/discrete-rational-spectrum-and-odometers Comment by FelipeG FelipeG 2012-07-11T22:02:39Z 2012-07-11T22:02:39Z I think that works, thanks. http://mathoverflow.net/questions/87952/a-system-of-equations-for-integers/87995#87995 Comment by FelipeG FelipeG 2012-02-10T18:10:31Z 2012-02-10T18:10:31Z Yes , that exactly what I meant. by &quot;Given R1, the system has a unique solution.&quot; thanks for clearing this out. http://mathoverflow.net/questions/87952/a-system-of-equations-for-integers Comment by FelipeG FelipeG 2012-02-10T17:42:10Z 2012-02-10T17:42:10Z Because I want things like $R_{m-1}C_{2}$ to be zero.