[FOM] Query concerning measure.
Bill Taylor
W.Taylor at math.canterbury.ac.nz
Mon Feb 20 23:49:55 EST 2006
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.
[ #4 is not exactly a property of *measures* specifically, OC! ]
He observed that "it has turned out" that we can have any THREE of these
that we want, but not all FOUR. A very intriguing little bagatelle
that has stuck in my mind all these years!
He told us that modern math keeps 2,3,4 and drops 1; and so we went
on to Lebesgue measure and so forth.
......
But I have wondered, on and off over the years, what about the other
three cases of these 3/4-option possibilities?
The only one that I could never seem to find out about was as in
my query - can one obtain 134 easily? And it seems one can. Great.
That was all way back in 1965. (Damaging admissions!!)
A historical question arises out of this - from what I can gather
my lecturer was being right up to the minute! Because it seems that
the 123 option had only been confirmed that very same year - is that so?
Indeed, if anyone can provide definitive dates for the discovery
of each of these options - when they became known to be consistent,
that would be cool. The oldest seems to be the "standard option",
234, which presumably became OK as soon as Lebesgue (or Borel?)
produced their measure.
So, can anyone supply the four dates please?
Bill Taylor
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