Timeline for Is this a closed set?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 21, 2013 at 7:47 | answer | added | Pietro Majer | timeline score: 3 | |
| Oct 21, 2013 at 2:39 | history | edited | Tom LaGatta | CC BY-SA 3.0 |
added 215 characters in body
|
| Oct 21, 2013 at 1:33 | comment | added | Tom LaGatta | @RickyDemer: yes, I made the implicit assumption that the probability measures are all Borel, and maybe even Radon. If the support doesn't exist (or is empty), then my question is vacuous as stated. However, Vakhania introduced an alternate notion of support, which is always well-defined. When the topological support exists, the two notions agree, and the Vakhania support has full measure. See Vakhania's classic article here: projecteuclid.org/DPubS/Repository/1.0/… | |
| Oct 21, 2013 at 1:25 | comment | added | user5810 | Are your probability measures on $X$'s Borel sets? $\:$ What if there is no smallest closed set of full measure? | |
| Oct 21, 2013 at 0:50 | comment | added | fedja | Certainly not closed: take $\Theta=[0,1]$, $X=\mathbb R$, $P_0=\delta_0$ and $P_\theta=\frac{\theta}{\pi(x^2+\theta^2)}\,dx$ for $\theta>0$ (or any other approximate identity). I'm not sure about Borel measurability: need to think more. | |
| Oct 21, 2013 at 0:44 | history | edited | Tom LaGatta | CC BY-SA 3.0 |
edited title
|
| Oct 21, 2013 at 0:25 | comment | added | Tom LaGatta | Here's the naïve approach, which doesn't work. Suppose $(\theta^s, x^s) \to (\theta,x)$. Let $U$ be an open neighborhood of $x$ in $X$. We want to show that $\mathbb P_\theta(U) > 0$. By weak convergence of measures, $\mathbb P_\theta(U) \le \liminf_s \mathbb P_{\theta^s}(U)$. Unfortunately, this is the wrong direction for the inequality, and doesn't show that $\mathbb P_\theta(U)$ is positive. | |
| Oct 21, 2013 at 0:11 | history | asked | Tom LaGatta | CC BY-SA 3.0 |