[FOM] Category and Measure
W.Taylor at math.canterbury.ac.nz
Fri Sep 21 02:17:06 EDT 2007
J Shipman writes:
> There is something very mysterious going on here, which leads me
> to suspect that one or the other of the two notions of "typical"
> (co-null or co-meager) is ultimately more fundamental and applicable
> in "real life" than the other.
This is a very intriguing matter, which I hope some expert folk
will respond to.
> For my money, measure seems a more robust concept than category
> as far as applications to physics are concerned,
That has been my impression also; and I suspect that of most of us.
It may be due to the fact that measure has "degrees" between null and full.
Subsets of [0,1] might have any measure between 0 and 1 inclusive.
This seems to give it more force.
> but I'd be happy to hear an opposing view.
Not really an opposing view, but on further refelction I suspect my
"degrees of" observation just above, may be somewhat superficial.
Lebesgue himself proved his density theorem - essentially that any set's
measure-density very close to a single point (in a standard sense)
could be only 0 or 1, and nothing in between. So e.g. one cannot have
a subset of [0,1] which is of relative measure 1/2 in every sub-interval.
This makes measure look much more like category.
And it leads to the question - which is the main point of my response -
can a similar thing be done with category?
That is, we know a subset of [0,1] might have "null category" (meagre), or
"full category" (co-meagre); but can things in between be sensibly defined?
Has this been done anywhere? That is, can one identify the sub-intervals
of meagreness and fullness, and add up their respective lengths,
the latter being the "amount of category" in the original subset?
Just as one can do now for measure, I think.
Of course not every subset will be "categorizable", just as not every
subset is measurable. But it seems doable with at least some simple
So, has it all been done anywhere?
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