Can an uncountable family of positive-measure sets be such that no point belongs to uncountably many of them?

Question

I would be happy to know whether the following is true:

For every uncountable family $\Gamma$ of positive-measure sets in a $\sigma$-finite measure space, there is at least one point that belongs to uncountably many members of $\Gamma$.

And if this is false for general $\sigma$-finite measure spaces, is it true for Lebesgue measure?

Asaf Karagila · Answer 1 · 2013-03-03T11:48:58.597

7

Here is a quick counterexample under the assumption of CH:

Let $\Bbb R=\{r_\alpha\mid\alpha<\omega_1\}$, and let $A_\alpha=\{r_\beta\mid\alpha<\beta<\omega_1\}$. All those are cocountable therefore certainly have a full [Lebesgue] measure.

But if $x\in\Bbb R$ then $x=r_\alpha$ for some $\alpha$ and so $x\notin A_\beta$ for any $\beta>\alpha$, so it only appears in a countable number of the sets.

edited Mar 03 '13 at 11:48

answered Mar 03 '13 at 10:58

Asaf Karagila

393,674

Is this a $\sigma$-finite measure space, as requested? – Gerry Myerson Mar 03 '13 at 11:31
@Gerry: Is the Lebesgue measure sigma finite? – Asaf Karagila Mar 03 '13 at 11:48
@Asaf: Thanks, that is super helpful! – Cian Mar 03 '13 at 11:56
However I am still curious about how the conjecture fares on the assumption that the continuum hypothesis is false... – Cian Mar 03 '13 at 11:56
@Cian Dorr: I feel that it should be probably false, but I am not going to fall off my chair if it is consistently false. Maybe under MA+$\lnot$CH... I don't know now. I'll think about it some more. – Asaf Karagila Mar 03 '13 at 11:59
Asaf, I don't know, it has been a long time since I've looked at these things, and I may never have looked at them with $\omega_1$ involved. – Gerry Myerson Mar 03 '13 at 12:12
@Gerry: Well, CH has its applications in measure theory. – Asaf Karagila Mar 03 '13 at 12:25
@Gerry: Just to be clear, Lebesgue measure certainly is $\sigma$-finite - the real line is the union of countably many finite-measure sets ..., [-1,0],[0,1],.... – Cian Mar 03 '13 at 12:30
@Cian: I am starting to feel as if this is independent of ZFC. – Asaf Karagila Mar 03 '13 at 12:37
OK, I was aware that Lebesgue measure is $\sigma$-finite on the real line, it's the ${{r_{\alpha}\mid\alpha\lt\omega_1}}$ that had me going. – Gerry Myerson Mar 03 '13 at 22:36
The other direction follows from MA + not CH since every ccc poset of size $\omega_1$ is sigma centered. A more direct way to do this would be to add $\omega_2$ random reals. – hot_queen Dec 12 '15 at 03:57

Can an uncountable family of positive-measure sets be such that no point belongs to uncountably many of them?

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