Timeline for Example of measure for some algebra over N
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 8, 2019 at 8:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 5, 2018 at 20:45 | comment | added | YCor | If you consider the counting measure on $\mathbf{Z}/k\mathbf{Z}$ and consider its inverse image on $\mathbf{Z}$ (defined on the $\sigma$-algebra $\mathcal{A}$ consisting of subsets invariants by $\pm k$), then you get such a measure, but on $\mathbf{Z}$. After intersecting with $\mathbf{N}$, you get a measure almost as required, except the last axiom. Indeed, writing $T(n)=n+1$, your last axiom is that $A$ measurable implies $T(A)$ measurable and $\mu(T(A))=\mu(A)$. What holds here is that $A$ measurable implies $T^{-1}(A)$ measurable and $\mu(T^{-1}(A))=\mu(A)$. | |
| Dec 5, 2018 at 3:50 | answer | added | user44191 | timeline score: 1 | |
| Dec 4, 2018 at 0:18 | comment | added | Yemon Choi | Are you requiring $\mu({\bf N})=1$, i.e.\ some kind of left-invariant mean? | |
| Dec 3, 2018 at 11:15 | review | Close votes | |||
| Dec 6, 2018 at 20:05 | |||||
| Dec 3, 2018 at 10:10 | comment | added | user44191 | There is only one translation-invariant countably additive measure on $\mathbb{N}$, up to scaling: the counting measure. Additionally, why not just take $\mathcal{A}$ as the full $\sigma$-algebra itself? | |
| Dec 3, 2018 at 9:24 | answer | added | Dominic van der Zypen | timeline score: 0 | |
| Dec 3, 2018 at 9:14 | history | edited | Lisa | CC BY-SA 4.0 |
deleted 8 characters in body
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| Dec 3, 2018 at 9:10 | history | edited | user44191 | CC BY-SA 4.0 |
Cleared up grammatical issues
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| Dec 3, 2018 at 9:00 | review | First posts | |||
| Dec 3, 2018 at 9:43 | |||||
| Dec 3, 2018 at 8:59 | history | asked | Lisa | CC BY-SA 4.0 |