[FOM] Category and Measure

Bill Taylor 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
example subsets.

So, has it all been done anywhere?

Bill Taylor

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