Measures beyond Lebesgue: are Solovay's proofs extendible to them?

[nomadreid]

In 1970, Solovay proved that,
although
(1) under the assumptions of ZF & "there exists a real-valued measurable cardinal", one could construct a measure μ (specifically, a countably additive extension of Lebesgue measure) such that all sets of real numbers were measurable (wrt μ),
nonetheless
(2) under the assumption of ZFC, one can construct a set (e.g., the Vitali set) which is not Lebesgue measurable.

However, I am not sure whether these proofs carry over to all measures: in other words, is it conceivable that, under ZFC & a sufficiently strong large cardinal axiom, there is a measure M so that all sets of real numbers are measurable wrt M? (For example, it would seem reasonable that the Vitali set is also not measurable by the μ in (1), but what of other measures?)

 

[mathman]

I am not familiar with the work you are discussing. However with a measure, where each point has measure 1, all sets are measurable.

 

[pwsnafu]

However, I am not sure whether these proofs carry over to all measures: in other words, is it conceivable that, under ZFC & a sufficiently strong large cardinal axiom, there is a measure M so that all sets of real numbers are measurable wrt M? (For example, it would seem reasonable that the Vitali set is also not measurable by the μ in (1), but what of other measures?)

The trivial measure (sends every set to zero) does the job.

If you want non-trivial example consider the Dirac measure.

 

[nomadreid]

Ah, thanks to both of you, mathman and pwsnafu. My mistake was that I was unconsciously thinking only of additive measures, so that I simply overlooked the obvious ones you mentioned.
If I had put additive measure as a prerequisite, as well as translation invariant, I would have essentially ended up with Lebesgue measure, which apparently would have made my question vacuous.

 

[blue_raver22]

I am intrigued by the question of whether Solovay's proofs can be extended to measures beyond Lebesgue. While his work has been groundbreaking in our understanding of measurable sets and measures, it is always important to continue exploring and pushing the boundaries of our knowledge.

There have been some attempts to extend Solovay's proofs to other measures, such as the Baire measure and Hausdorff measure. However, these attempts have not been successful in fully replicating Solovay's results. This suggests that there may be some fundamental differences between Lebesgue measure and other measures that make it uniquely suited for Solovay's proofs.

One possible avenue for further exploration is to consider the properties of the measures themselves and how they relate to the properties of the sets being measured. It is possible that certain measures may have different properties that make them more amenable to Solovay's proofs.

Another important consideration is the role of large cardinal axioms in Solovay's proofs. While these axioms have been crucial in establishing the existence of certain measures, it is possible that they may also play a role in determining the scope of Solovay's proofs. Further research in this area may shed light on whether these proofs can be extended to measures beyond Lebesgue.

Overall, the question of whether Solovay's proofs can be extended to measures beyond Lebesgue is an important one that requires further investigation. As scientists, it is our duty to continue exploring and expanding our understanding of the mathematical world, and this question presents an exciting opportunity to do so.