Timeline for Finite measure on the power set
Current License: CC BY-SA 3.0
11 events
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| Nov 2, 2013 at 1:40 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
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| Jul 31, 2012 at 14:50 | comment | added | Andrés E. Caicedo | Xander: Yes, $W$ was a typo for $E$. Fixed now. Thanks. It may perhaps be worth pointing out two remarks: 1. If there is a real-valued measurable $\kappa$, then any $X$ with $|X|\ge\kappa$ admits such a measure: Simply concentrate it on subsets of $Y$, where $Y$ is a subset of $X$ of size $\kappa$; and on subsets of $Y$ simply assign a measure via a bijection with $\kappa$. 2. In fact, if there is an atomless real-valued measurable, and $X=\mathbb R$, we can find a measure on all subsets of $X$ that extends Lebesgue measure. | |
| Jul 31, 2012 at 14:46 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
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| Jul 31, 2012 at 8:38 | comment | added | Xander Faber | @Stefan - That helps a great deal, at least morally, for what I'm trying to do. Thanks. | |
| Jul 31, 2012 at 8:37 | vote | accept | Xander Faber | ||
| Jul 31, 2012 at 8:08 | comment | added | Stefan Geschke | I should add to my comment that the reason Andres mentions measurable cardinals is that their equiconsistency with real valued measurables shows that you cannot construct a $\sigma$-additive measure on $\mathcal P(\mathbb R)$ without the help of some strong additional axioms. | |
| Jul 31, 2012 at 8:02 | comment | added | Stefan Geschke | Xander, measurable cardinals are much bigger than $\mathbb R$. It is the so-called real valued measurables that could possibly be $\leq|\mathbb R|$ and that are connected to measures on $\mathcal P(\mathbb R)$. | |
| Jul 31, 2012 at 7:17 | comment | added | Xander Faber | I don't have any sense of whether or not a given set $X$ can be put in bijection with a measurable cardinal (again identifying cardinals with sets of ordinals). So for example, is it known when $X = \mathbb{R}$? I assume this case was the original motivation for the question. | |
| Jul 31, 2012 at 7:00 | comment | added | Xander Faber | Ack! This is the second time in two years that I've accidentally bumped into some serious set theory / foundations in my research. @Andres - Thanks for putting a name to this question and pointing me to some literature. And yes, I did mean $\sigma$-additive. Is $W = E$ in your third paragraph, line 6? | |
| Jul 31, 2012 at 6:56 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
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| Jul 31, 2012 at 6:07 | history | answered | Andrés E. Caicedo | CC BY-SA 3.0 |