Real-valued measurable cardinals versus Vitali sets

[nomadreid]

If there exists a real-valued measurable cardinal, then there is a countably additive extension of Lebesgue measure to all sets of real numbers. This would include then the Vitali sets, which are an example of sets that are not Lebesgue measurable for weaker assumptions than the existence of a real-valued measureable cardinal. However, after going over the proof that a Vitali set is not measurable, for example in Wikipedia's "Vitali set", I do not see where the proof would fail under the assumption of a real-valued measureable, i.e., assuming that there exists a cardinal κ so that there is an atomless κ-additive measure on the power set of κ. I presume I am missing something breathtakingly obvious. Could someone point this out to me?

 

[AKG]

If there exists a real-valued measurable cardinal, then there is a countably additive extension of Lebesgue measure to all sets of real numbers. This would include then the Vitali sets, which are an example of sets that are not Lebesgue measurable for weaker assumptions than the existence of a real-valued measureable cardinal. However, after going over the proof that a Vitali set is not measurable, for example in Wikipedia's "Vitali set", I do not see where the proof would fail under the assumption of a real-valued measureable, i.e., assuming that there exists a cardinal κ so that there is an atomless κ-additive measure on the power set of κ. I presume I am missing something breathtakingly obvious. Could someone point this out to me?

The Vitali set proves that there exists no measure on all sets of reals which is:

1. countably additive
2. translation invariant, and
3. assigns [0,1] the measure 1

The existence of a real-valued measurable gives you a countably additive total extension of Lebesgue measure, and since Lebesgue measure assigns a measure of 1 to [0,1], this extension will too. So what must go wrong is that this extensions must fail to satisfy condition 2: translation invariance.

By the way your choice of words "breathtakingly obvious" was pretty amusing. I think you meant "painfully obvious," but then confused "painfully" with "painstakingly," and then "painstakingly" with "breathtakingly" :D

 

[nomadreid]

The Vitali set proves that there exists no measure on all sets of reals which is:
1. countably additive
2. translation invariant, and
3. assigns [0,1] the measure 1
The existence of a real-valued measurable gives you a countably additive total extension of Lebesgue measure, and since Lebesgue measure assigns a measure of 1 to [0,1], this extension will too. So what must go wrong is that this extensions must fail to satisfy condition 2: translation invariance.

Excellent. Thank you very much, AKG. That answers the question perfectly.

By the way your choice of words "breathtakingly obvious" was pretty amusing. I think you meant "painfully obvious," but then confused "painfully" with "painstakingly," and then "painstakingly" with "breathtakingly" :D

Actually, there was no confusion: I used this combination of words on purpose in order that its incongruity would emphasize the meaning, just as some people use words of bodily functions or religious entities to do the same. I was inspired by my favourite court judgement of all time, whereby a judge in the U.S. called the arguments of Intelligent Design proponents "breathtakingly inane." Anyway, I'm glad it was able to amuse.