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

  • Thread starter nomadreid
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  • #1
nomadreid
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Main Question or Discussion Point

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?)
 

Answers and Replies

  • #2
mathman
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I am not familiar with the work you are discussing. However with a measure, where each point has measure 1, all sets are measurable.
 
  • #3
pwsnafu
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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.
 
  • #4
nomadreid
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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.
 

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