User peter schmitt - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:04:23Z http://mathoverflow.net/feeds/user/1786 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10186/what-are-some-interesting-sequences-of-functions-for-thinking-about-types-of-conv/10209#10209 Answer by Peter Schmitt for What are some interesting sequences of functions for thinking about types of convergence? Peter Schmitt 2009-12-31T01:20:56Z 2010-01-02T11:45:32Z <p>On $[0,1]$: $f_n = a_n\chi_{[\alpha n,\alpha n + \varepsilon n^{-2}]\ {\rm mod}\ 1 }$ with $\alpha$ irrational, and $a_n = 1$ or $a_n = n^2$.</p> <p>This is, of course, also similar ...</p> <p>Transferred from my comments below, and corrected (TeX was not shown, so I did not see that some of the code did not work):</p> <p>Well, it was rather late (in my timezone). So I only typed in the $f_n$. </p> <p>$\int |f_n -0|$ is either $= \varepsilon n^{-2} \to 0$ or $= \varepsilon \to \varepsilon$, respectively. </p> <p>$\int \bigcup_{n\ge N} \lbrace x | \chi_{[\alpha n,\alpha n + \varepsilon n^{-2}]\ {\rm mod}\ 1 } \ne 0 \rbrace \le \sum_{n\ge N}\varepsilon n^{-2} \to 0$, i.e., $f_n \to 0$ almost everywhere.</p> http://mathoverflow.net/questions/5351/whats-an-example-of-a-space-that-needs-the-hahn-banach-theorem/5471#5471 Answer by Peter Schmitt for What's an example of a space that needs the Hahn-Banach Theorem? Peter Schmitt 2009-11-14T01:49:37Z 2009-11-14T01:49:37Z <p>The simplest example I know are the real numbers as vector space over the rationals. The Hahn-Banach theorem asserts the existence of additive functionals other than standard addition, For instance, $f$ with $f(1)=1, f(\pi)=0$.</p> http://mathoverflow.net/questions/10186/what-are-some-interesting-sequences-of-functions-for-thinking-about-types-of-conv/10209#10209 Comment by Peter Schmitt Peter Schmitt 2009-12-31T11:27:29Z 2009-12-31T11:27:29Z How do I see the last comment TeXed? If I use &quot;add / show 2 more comments&quot; only raw TeX is shown? http://mathoverflow.net/questions/10186/what-are-some-interesting-sequences-of-functions-for-thinking-about-types-of-conv/10209#10209 Comment by Peter Schmitt Peter Schmitt 2009-12-31T11:10:39Z 2009-12-31T11:10:39Z It seems I do not yet completely understand editing here. (First it did not show the formulas) So I overlooked the mistake - it should read: $\int \bigcup_{n\ge N} \{ x | \chi_{[\alpha n,\alpha n + \varepsilon n^{-2}]\ {\rm mod}\ 1 } \not= 0 \} \le \sum{n\ge N}\varepsilon n^{-2}$ \to 0 $http://mathoverflow.net/questions/10186/what-are-some-interesting-sequences-of-functions-for-thinking-about-types-of-conv/10209#10209 Comment by Peter Schmitt Peter Schmitt 2009-12-31T10:46:43Z 2009-12-31T10:46:43Z Is there no TeX available for comments? http://mathoverflow.net/questions/10186/what-are-some-interesting-sequences-of-functions-for-thinking-about-types-of-conv/10209#10209 Comment by Peter Schmitt Peter Schmitt 2009-12-31T10:43:59Z 2009-12-31T10:43:59Z Well, it was rather late (in my timezone). So I only typed in the$f_n$.$ \int f_n -0 $is$ = \varepsilon n^{-2} $or$ = \varepsilon $, respectively.$ \int \bigcup_{n\ge N} \{ x | \chi_{[\alpha n,\alpha n + \varepsilon n^{-2}]\ {\rm mod}\ 1 } \not= 0 \} \le 1 - \sum{n\ge N}\varepsilon n^{-2}$\to 1$