Timeline for Is every $\sigma$-algebra generated by some measurable function? [closed]
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2023 at 12:45 | comment | added | Joel David Hamkins | That $\sigma$-algebra is not countably generated (a fun exercise), and so by Gerald's observation this is not generated by any measurable function via the Borel sets, which are countably generated. | |
| Nov 14, 2023 at 18:55 | history | closed |
Jochen Wengenroth Gerald Edgar Christian Remling Max Horn Tobias Fritz |
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| Nov 14, 2023 at 16:49 | comment | added | MatEZ | I am asking because I read, that $\mathcal{C}:=\{ C\subset \mathbb{R}: C\text{ Is countable or } C^c \text{ is countable}\}$ Is not generated by any function $f:\mathbb{R}\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$ And I didn't believe it. Thanks for clarification | |
| Nov 14, 2023 at 16:48 | review | Close votes | |||
| Nov 14, 2023 at 18:55 | |||||
| Nov 14, 2023 at 16:43 | comment | added | Gerald Edgar | This is the wrong forum for the question..... Answer: YES, in the case: real-valued measurable function $f$ and countably-generated $\sigma$-algebra $\mathcal A$. But otherwise, perhaps not. So, if $\mathcal B$ is countably generated, then so is $\sigma(f)$. | |
| Nov 14, 2023 at 16:29 | comment | added | MatEZ | And if I consider $|\mathcal{A}|\leq|\mathcal{B}|$? | |
| Nov 14, 2023 at 16:28 | comment | added | Jochen Wengenroth | Of course not. For $X=\{0,1\}$ and $\mathcal B$ the power set, every $\sigma$-algebra of $\sigma(f)$ has at most four elements. | |
| Nov 14, 2023 at 16:24 | comment | added | MatEZ | Given measurable spaces $(X,\mathcal{B})$ and $(\Omega,\mathcal{A})$, does there exists a function $f:\Omega\rightarrow(X,\mathcal{B})$ such that $f$ generates $\mathcal{A}$? | |
| Nov 14, 2023 at 16:20 | comment | added | Joel David Hamkins | Can you clarify the quantifiers? You are asking whether there is such a measurable space $(X,\mathcal{B})$, which will generate every $(\Omega,\mathcal{A})$ via some measurable $f$? (Also, what is a "general measurable space"?) | |
| Nov 14, 2023 at 16:13 | history | asked | MatEZ | CC BY-SA 4.0 |