User alberto santini - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:24:17Z http://mathoverflow.net/feeds/user/12563 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58554/preferably-rare-audio-video-recordings-of-famous-mathematicians/58560#58560 Answer by Alberto Santini for (Preferably rare) Audio/Video recordings of famous mathematicians? Alberto Santini 2011-03-15T18:54:41Z 2011-03-15T18:54:41Z <p>On YouTube there are a lot from <a href="http://www.youtube.com/results?search_query=paul+erdos" rel="nofollow">Paul ErdÅ‘s</a>. Not to mention the wonderful movie <a href="http://www.zalafilms.com/films/nisanumber.html" rel="nofollow">N is a number</a>.</p> http://mathoverflow.net/questions/56925/probability-measures-and-cardinality-c/56927#56927 Answer by Alberto Santini for Probability Measures and Cardinality > c Alberto Santini 2011-02-28T21:51:44Z 2011-02-28T21:56:47Z <p>There's even more. If your set can be considered as a product of probability spaces, its measure will have some good properties. This is a theorem due to Alexandra Ionescu Tulcea: given an infinite (with arbitrary cardinality) family of probability spaces $$\left\{(\Omega_n, \mathscr{B}_n, \mu_n)\right\}_{n \in I}$$ if we call $$\pi_m : \prod_{n \in I} \Omega_n \rightarrow \Omega_m$$ the canonical projection, then it's possible to give $\Omega \mathrel{\mathop{:}=} \prod_{n \in I} \Omega_n$ the structure of a probability space with measurable sets $$\mathscr{B} = \bigotimes_{n \in I} \mathscr{B}_n$$ and probability measure $\mu : \mathscr{B} \rightarrow \mathbb{R}$ such that $$\forall {n_1, \ldots, n_t} \subset I \ \ \forall (A_{n_1}, \ldots, A_{n_t}) \in \prod_{i = 1}^t \mathscr{B}_{n_i}$$ $$\mu\left(\pi_{n_1}^{-1}(A_{n_1}) \cap \ldots \cap \pi_{n_t}^{-1}(A_{n_t})\right) = \mu_{i_1}(A_{i_1}) \ldots \mu_{i_t}(A_{i_t})$$</p> http://mathoverflow.net/questions/45784/does-pointwise-convergence-imply-uniform-convergence-on-a-large-subset/56783#56783 Answer by Alberto Santini for Does pointwise convergence imply uniform convergence on a large subset? Alberto Santini 2011-02-27T03:12:11Z 2011-02-27T03:12:11Z <p>Also, <a href="http://www.projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.nmj/1118798877" rel="nofollow">Shinoda (1973)</a> proved that if Martin's Axiom and $\neg CH$ hold, then the answer to 1) si affirmative.</p>