Concept of measure in undergraduate mathematics.

From: Jim Spriggs (jim.sprigs_at_ANTISPAMbtinternet.com.invalid)
Date: 03/21/05 

Date: Mon, 21 Mar 2005 22:11:04 +0000 (UTC)

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? 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:

  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?

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?

[*] 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.

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