Re: motivation for definition of Lebesgue-measurable set

In article <180120071227275336%edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
G. A. Edgar <edgar@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
In article <1169135672.883740.275080@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
hagman <google@xxxxxxxxxxxxx> wrote:


This method seems to go back to Carathodory.
I prefer Lebesgue-measurable to mean lying between two Borel sets of
equal measure.

Unfortunately, that is no good for sets of infinite measure.

Caratheodory used his definition to extend the notions to
non-sigma-finite situations, like "arc length" in the plane.

For non-sigma-finite situations, this gives different results
than other extensions, and I find this one both non-intuitive
and giving the wrong results. In the case of sigma finite,
it is equivalent to, for every epsilon > 0, the set differing
from a countable union of elements of the field by less than
a countable union of elements of the field, the sum of whose
measures is less than epsilon. This latter works in all
sigma-finite cases, gives the extension, and can be easily
modified to handle the "good" non-sigma-finite cases. It is
also easy to prove the theorems.

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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