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 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 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 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 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 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 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 $