- #1

nomadreid

Gold Member

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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?)

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?)