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I have an intuition that most subsets of reals are not Lebesgue measurable. The tricky part is defining "most". For that, we need a measure on the set of subsets of the powerset of reals, in other words, a measure on the powerset of the powerset of reals. Has anyone defined such a measure, and either confirmed or denied my intuition?

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    $\begingroup$ Here's one silly observation via category rather than measure. Consider the function $$d:\mathcal{P}(\mathbb{R})\times\mathcal{P}(\mathbb{R})\rightarrow \mathbb{R}: (X,Y)\mapsto \mu^*(X\triangle Y),$$ where $\mu^*$ is Lebesgue outer measure and $\triangle$ is symmetric difference. This is a pseudometric since we always have $$(X\triangle Y)\cup(Y\triangle Z)\supseteq X\triangle Z$$ and outer measure is finitely subadditive. In the topology coming from this pseudometric, the set of measurable sets is nowhere dense. So "generic sets aren't measurable." $\endgroup$ Jul 28, 2021 at 6:54
  • $\begingroup$ @Noah Schweber: Your comment should be an answer. In fact, I just cited it here. $\endgroup$ Jul 28, 2021 at 16:38
  • $\begingroup$ @DaveL.Renfro I mean, the OP specifically asks for a measure-based approach - I feel like category is far enough from measure that (until/unless they modify the question) it doesn't really count. $\endgroup$ Jul 28, 2021 at 17:02
  • $\begingroup$ @DaveL.Renfro In retrospect I'm probably being too strict. Added! $\endgroup$ Jul 28, 2021 at 17:51

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The following isn't entirely an answer, since you specifically ask about measure and this is an observation about category, but I've come around to think that it's more than a comment (which is what it originally was).


Consider the function

$$d:\mathcal{P}(\mathbb{R})^2\rightarrow\mathbb{R}:(X,Y)\mapsto\mu^*(X\triangle Y),$$ where $\mu^*$ is Lebesgue outer measure and $\triangle$ is symmetric difference. This is a pseudometric since we always have $$(X\triangle Y)\cup (Y\triangle Z)\supseteq X\triangle Z$$ and outer measure is finitely subadditive. In the topology coming from this pseudometric, the set of measurable sets is nowhere dense. So "generic sets aren't measurable." (Separately it's worth noting that measurability is well-defined up to distance zero: if $d(X,Y)=0$ and $X$ is measurable then $Y$ is measurable, since all null sets are measurable. So if we "mod out by distance zero" to get a genuine metric space, we don't lose any information as far as measurability is concerned - which I think helps emphasize the naturality of this approach.)

Of course generic is not the same as random, and I don't see a similar way to put a measure on $\mathcal{P}(\mathbb{R})$ (or anything like a measure) which would help in this context.

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A more in-depth approach to Noah’s Answer has been attempted in the paper cited below (there’s no free link to the paper, but perhaps one could ask Dave L. Renfro):

According to Dave:

Let ${\mathbb P}[0,1]$ be the collection of all subsets of $[0,1]$ modulo the equivalence relation $\sim$ defined by $E \sim F \Leftrightarrow {\lambda^{*}(E \Delta F)} = 0,$ where $\lambda^{*}$ is Lebesgue outer measure and $\Delta$ is the symmetric difference operation on sets. The set ${\mathbb P}[0,1]$ can be made into a complete metric space by defining the distance function $d,$ where $d(E,F) = {\lambda^{*} (E \Delta F)}.$ In the paper cited below it is proved that the collection of measurable subsets of $[0,1]$ is a perfect nowhere dense set in ${\mathbb P}[0,1].$

Thus, in this setting, the collection of measurable subsets of $[0,1]$ makes up a very tiny part of the collection of all the subsets of $[0,1].$ Note that in the space ${\mathbb P}[0,1]$ the collection of measurable subsets is not just a first category subset of ${\mathbb P}[0,1]$ (this alone would make the collection a tiny subset of ${\mathbb P}[0,1]$), but in fact the collection of measurable subsets is actually a nowhere dense subset of ${\mathbb P}[0,1]$ (hence my saying the collection is a very tiny subset of ${\mathbb P}[0,1]$).

Nobuyuki Kato, Tadashi Kanzo, and Oharu Shinnosuke, A note on the measure problem, International Journal of Mathematical Education in Science and Technology 19 (1988), 315-318.

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