[FOM] Query concerning measure.

[Robert M. Solovay]

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 
measurable'.

 	--Bob Solovay

>