[FOM] Query concerning measure.
Robert M. Solovay
solovay at Math.Berkeley.EDU
Tue Feb 21 04:33:36 EST 2006
On Tue, 21 Feb 2006, Bill Taylor wrote:
> Many thanks to those who responded to my query (both on and off list).
> It has resolved an old puzzle of mine.
> It may have looked an odd query to spring out of the blue, so I thought
> I'd give an account of what lies behind it. It may be OT for the list
> but let's give it a try.
> Many years ago, my final honours lecturer in measure theory made
> a curious little list on the blackboard, which I'm pretty sure I've
> recalled correctly, but is not in my old notes, (annoyingly!)
> He was speaking on the matter of extending the concept of length,
> to more general sets of reals than mere intervals. He wrote up three
> (or four) properties of such a pre-measure that we would like to keep.
> 1. All sets become measurable.
> 2. It is countably additive, not merely finitely.
> 3. It is translation invariant.
> 4. Axiom of Choice.
I did 123 in 1964 [published much later] and 124 in 1967. The
first requires an inaccessible; the second a measurable cardinal. So your
teacher was prescient.
Pedantic footnote. Eg for 123 I showed "if there is a model of ZFC + an
inacessible" then there is a model of "ZF + DC + 'All sets Lebesgue
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