Timeline for Lebesgue outer measure
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 8, 2016 at 8:17 | comment | added | Taras Banakh | The problem of @Lebesgue is equivalent to the following one: Is there an atomless Borel probability measure $\mu$ on a metrizable separable space $X$ such that each subset $A$ of $X$ is $\mu$-measurable (in the sense that $B_1\subset A\subset B_2$ for some Borel subsets $B_1,B_2$ of $X$ such that $\mu(B_1)=\mu(B_2)$). I have a strong feeling that such a problem could not be open. Maybe Fremlin in his fundamental "Measure Theory" writes something essential on this topic? | |
| Dec 8, 2016 at 8:12 | comment | added | Taras Banakh | A natural attempt to construct a (consistent) example of a set $X$ with $\mu^*|\mathcal P(X)$ being a measure is to find a non-measurable subset $X$ of the real line such that all subsets of $X$ are Borel in $X$. It is not clear if such a set can exist at all: by a result of Fletcher (1978), mentioned in the Miller;s survey "Special subsets of the real line" a subset $X$ of the real line has is Lebesgue null if all subsets of $X$ are relative $F_\sigma$-sets in $X$. | |
| Dec 6, 2016 at 22:51 | history | edited | Taras Banakh | CC BY-SA 3.0 |
A negative answer under $non(L)=c$ is given.
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| Dec 6, 2016 at 22:48 | comment | added | Taras Banakh | I added a proof under $non(L)=c$. But ZFC-question is still not resolved. It would be interesting to understand what happens for a set $X$ of cardinality $|X|=non(L)<\mathfrak c$ with $\mu^*(X)>0$. | |
| Dec 6, 2016 at 22:45 | history | edited | Taras Banakh | CC BY-SA 3.0 |
A negative answer under $non(L)=c$ is given.
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| Dec 6, 2016 at 0:16 | comment | added | Nate Eldredge | But isn't it also consistent that the continuum is real-valued measurable? So this doesn't resolve the question in ZFC. | |
| Dec 5, 2016 at 21:38 | history | answered | Taras Banakh | CC BY-SA 3.0 |