Re: Concept of measure in undergraduate mathematics.
From: Herman Rubin (hrubin_at_odds.stat.purdue.edu)
Date: 03/22/05
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Date: 22 Mar 2005 12:46:09 -0500
In article <423F467A.7846B102@ANTISPAMbtinternet.com.invalid>,
Jim Spriggs <jim.sprigs@ANTISPAMbtinternet.com.invalid> wrote:
>The thread "Origins of analysis" is dealing with (among other things)
>the problem of understanding analysis with or without limits, and
>understanding limits with or without deltas and epsilons. When I did my
>degree, right at the start there was a course on real analysis that
>began with real numbers and Dedekind cuts and then went on to limits in
>the delta/epsilon style. And I understood it. Not only did I
>understand it, I actually enjoyed seeing how the calculus of my school
>days was made rigorous [*]. Also, right at the start, there was a
>course on probability which I never got to grips with. Not only could I
>make no sense of the frequency "definition", I could see no connection
>between the discrete and the continuous cases. Later I read Kingman and
>Taylor's Introduction to Measure and Probability and all was made clear.
>My question is, could it not have been made clear from the start by
>defining probability in terms of measure?
One cannot really define probability. However, any
particular probability situation can be realized within
measure, and often in a wide variety of ways. It is
both useful and conceptually important to allow a large
number of these representations.
But you make the concept of measure much to difficult and
even too restricted. It is a MAJOR source of confusion to
start with measure on the real line. It is not clear
exactly how old the original idea, unfortunately messed up
by those who did not recognize the simplicity when they
started confusing antidifferentiation and integration, but
the earliest measure used was "number of elements". There
are lots of others. Also, the earliest use of integration
was computing the amount of a merchant's bill. The real
line as a basic space for measure hides the essentials.
The difference between the Riemann type (notice I did not
say Riemann) and the Lebesgue type (likewise) is most
easily seen if one starts integration with discrete
measures. A Riemann extension is one in which a function
is approximated by one taking a finite number of values,
and the integral is the limit of the corresponding finite
evaluations of integrals with respect to measures on finite
algebras of sets. A Lebesgue extension requires the measure
to be countably additive, and allows countable approximations.
Expectation can be used in the finitely additive case, and
much can be done there, if one is careful. Even the Fubini
Theorem has its version, but unfortunately the very important
Radon-Nikodym Theorem does not.
Furthermore, armed with the
>concept of measure, the student could go on and learn the Lebesgue
>integral and save time by not learning the Riemann integral. So I
>envisage a combined real analysis and probability course like this:
See the above. But real numbers and limits belong in
elementary school, if one is not too pedantic about them
and concentrates on ideas, not memorization, nor just
theorems and proofs. In fact, before your part I, the
integers, including induction, are needed.
> Analysis part I: real numbers, limits, sequences, series,
> derivatives.
> Probability part I: measure, measures P with P(whole space) = 1,
> probability.
> Analysis part II: Lebesgue integration.
> Probability part II: expectation, ... .
>Has anybody ever taught an _introductory_ course in probability in which
>probability was defined in terms of measure from the beginning? Has
>anybody ever considered introducing real analysis and probability in the
>above interleaved manner?
Any decent probability course does it, even at the very
definitely precalculus level. But not as you see it.
To understand the concepts of measure and integral, getting
them intertwined with the real line, as is usually done,
leaves great confusion.
>Another question: if the preferred manner of introducing Lebesgue
>integration is via the area under the graph of a step function (in the
>style of Weir's Lebesgue Integration and Measure, for example) could
>such an approach be used in something like the above scheme?
This is done too often. Lebesgue's original approach to
integration was better.
>[*] My memory may not be reliable, but I think school calculus
>(differential, at O level, at least) was done in terms of limits but the
>idea of a limit wasn't made very clear.
About the idea of a limit not being made clear, this is almost
universal. One can be quite rigorous and avoid the clumsy
formalism in the usual approaches; that does not make it very
clear, either.
-- 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@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
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