User simo_the_wolf - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:22:06Z http://mathoverflow.net/feeds/user/26825 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108240/existence-of-a-measure-under-certain-condition Existence of a measure under certain condition Simo_the_Wolf 2012-09-27T12:36:47Z 2012-09-30T16:12:35Z <p>Hi everyone,</p> <p>my problem seems quite simple: I have a set $\Gamma$ along with a nice $\sigma$-algebra $\mathscr{B}$. Then I have a vector space of bounded measurable functions $A \subset \mathscr{B}_{\infty} ( \Gamma )$, and a convex function $F : A \to [0, + \infty]$ which is lower semicontinuous with respect to pointwise convergence, and it satisfies:</p> <ul> <li>$F(0)=0$;</li> <li>$F( \lambda f ) = \lambda F( f)$ for every $f \geq 0$, $\lambda \geq 0$.</li> <li>$F( f) \geq \inf_{\Gamma} f$</li> </ul> <p>Can I find a nonnegative (sigma-additive) measure $\mathfrak{m}$ on $\Gamma$ such that $\int_{\Gamma} f d \mathfrak{m} \leq F(f)$ and $\mathfrak{m} (\Gamma) >0$? </p> <p>[This reminds me a bit Hahn-Banach theorem: recalling Daniell's integral I'm asking a linear functional $L$ that stays between $F$ and $\inf_{\Gamma} f$ (that is concave) and with additional condition that $L(f_n) \to 0$ everytime that I have a sequence $f_n$ decreasing to $0$.] </p> http://mathoverflow.net/questions/108330/when-is-the-graph-of-a-function-a-dense-set/108393#108393 Answer by Simo_the_Wolf for When is the graph of a function a dense set ? Simo_the_Wolf 2012-09-29T07:45:56Z 2012-09-29T07:45:56Z <p>Another example is any of the denegenerate solutions of the Cauchy functional equation <a href="http://en.wikipedia.org/wiki/Cauchy" rel="nofollow">http://en.wikipedia.org/wiki/Cauchy</a>'s_functional_equation#Properties_of_other_solutions , which has also the property of being measurable; unfortunately, I don't know when this condition is achieved, due to the not-so-easy construction of the function: maybe using the linearity one can show something, I don't know...</p> <p>However this class of function suggests a solution to your problem, for example $f(a+ \sqrt{2} b ) = a - \sqrt{2} b$ whenever $a,b \in \mathbb{Q}$ and $f \equiv 0$ out of $\mathbb{Q} ( \sqrt{2} )$. In this way you have a measurable function (because it is $0$ out of a countable set) and it has the property of the dense graph.</p> http://mathoverflow.net/questions/108232/examples-of-random-variable-x-independent-to-each-of-a-and-b-but-not-to-a-b/108242#108242 Answer by Simo_the_Wolf for examples of random variable X independent to each of A and B, but not to (A,B) Simo_the_Wolf 2012-09-27T12:45:16Z 2012-09-27T12:45:16Z <p>Let $A$ and $B$ be indipendent and uniformly distributed in $S^1$ (for example you can take $\Omega = S^1 \times S^1$ with the uniform probability and $A$ and $B$ as the two projection); then it is clear that $X=A+B$ is indipendent of $A$, it is indipendent of $B$, but it isn't indipendent of $(A,B)$.</p> <p>In this reasoning it's crucial that the uniform law on $S^1$ is traslation invariant.</p> http://mathoverflow.net/questions/110713/vitali-covering-theorem-for-arbitrary-subsets-of-doubling-metric-spaces Comment by Simo_the_Wolf Simo_the_Wolf 2012-10-26T00:23:20Z 2012-10-26T00:23:20Z if it can help, in doubling length spaces the boundaries of balls are always negligible http://mathoverflow.net/questions/109003/eigenvalues-of-the-sum-of-powers-of-a-matrix Comment by Simo_the_Wolf Simo_the_Wolf 2012-10-06T15:11:27Z 2012-10-06T15:11:27Z Take the basis that makes the matrix $A$ in Jordan form (say in the algebric closure of the field you are working with); than it is easy to see that 1) is true, and so also 2). http://mathoverflow.net/questions/108240/existence-of-a-measure-under-certain-condition/108480#108480 Comment by Simo_the_Wolf Simo_the_Wolf 2012-09-30T16:34:46Z 2012-09-30T16:34:46Z And yes, by the way I can look at $\Gamma$ as a topological space (actually also a metric space, if I want) and the $\sigma$-algebra is that of Borelian sets... http://mathoverflow.net/questions/108240/existence-of-a-measure-under-certain-condition/108480#108480 Comment by Simo_the_Wolf Simo_the_Wolf 2012-09-30T16:32:35Z 2012-09-30T16:32:35Z Thanks!! I put my question in general terms because I tought it was helpful, but in fact, in my &quot;real&quot; problem I have also the condition that $F( 1) = 1$ where $1(x) = 1$ for all $x \in \Gamma$. http://mathoverflow.net/questions/108340/approximating-a-convex-function-by-a-piecewise-linear-function/108367#108367 Comment by Simo_the_Wolf Simo_the_Wolf 2012-09-28T19:16:50Z 2012-09-28T19:16:50Z It seems to me that with the method of $\varepsilon - \delta$ with subdifferentials the bound should depend on the second derivative of $f$ instead of the first derivative, in pariticular getting $\varepsilon \leq C \| D^2f \| k^{ -2/n}$. Of course with this estimate one cannot include the example $f(x)=|x|$ http://mathoverflow.net/questions/108240/existence-of-a-measure-under-certain-condition Comment by Simo_the_Wolf Simo_the_Wolf 2012-09-28T14:43:22Z 2012-09-28T14:43:22Z To be more clear, an abstract version could be: giving a functional F like this on a space X, and a subspace $A \subseteq X^*$, where $X^*$ means the algebraic dual, what condition on $F$ should I have to get the existence of a nonzero element of $A$ that is less than or equal to $F$ ? http://mathoverflow.net/questions/108240/existence-of-a-measure-under-certain-condition Comment by Simo_the_Wolf Simo_the_Wolf 2012-09-28T14:37:20Z 2012-09-28T14:37:20Z Thanks, now I corrected the statement... Yes, I mean a sigma additive measure; I wrote positive instead of nonnegative meaning that I wanted $\mathfrak{m} (\Gamma ) &gt;0$ http://mathoverflow.net/questions/108232/examples-of-random-variable-x-independent-to-each-of-a-and-b-but-not-to-a-b/108242#108242 Comment by Simo_the_Wolf Simo_the_Wolf 2012-09-27T18:17:33Z 2012-09-27T18:17:33Z Sure... By the way, also this method could be generalized to $n$ variables. It seems to be true also for countable many variables... Think of this example: take $\{ X_i \}_{i \in \mathbb{N}}$ i.i.d. variables which take values in $\mathbb{Z} / 2 \mathbb{Z}$ uniformly. Then consider $Y_n = \sum_{ i \leq n} X_i$ (where the sum is also in $\mathbb{Z} / 2 \mathbb{Z}$), and then take $Z$ as the limit of the $Y_i$s along some non-principal ultrafilter. Then $Z$ is indipendent of any $A$ proper subset of $\{ X_i \}_{i \in \mathbb{N}}$ but clearly not indipendent from all of them.