NetNews Usenet Archive 1992 #20/NN_1992_20.iso/spool/sci/math/11290

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From: [e-mail] (Daniel E. Platt)
Subject: Re: Lebesgue integral (was: Couple of questions
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Date: Fri, 11 Sep 1992 20:31:56 GMT
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In article <[e-mail]>, [e-mail] (SCOTT I CHASE) writes:
|> In article <[e-mail]>, [e-mail] (Daniel E. Platt) writes...
|> >
|> >I think you would end up flat on your back.  The problem is that many (most)
|> >of the techniques revolving around Fourier series and integrals, completeness
|> >of a basis, etc, ultimately involve being able to evaluate 'improper' integrals
|> >as a limit of an integral of a sequence of functions.  They often look like
|> >Dirac-delta functions (the word 'function' is a misnomer, it is more like
|> >a limit of a family of functions) which just hides the complexity under
|> >some notation so that physicists don't have to worry about L^2(R^3).
|>
|> Physicists (not including mathematical physicists) deal with this issue in
|> a handwaving kind of way.  For example, take me.  I studied the theory of
|> distributions a little bit a while back.  I know that the tempered
|> distributions are an interesting class of objects for studying QM, because
|> they are in a well-defined sense the largest class of "potential wave-functions"
|>  on which the Fourier transform is well-defined.  This means, to me, that the
|> other distributions cannot be used for wave-functions because they so unphysical
|> as to have no normal way to define the momentum-space representation. But more
|> than this, I do not know.

Most physicists seem to believe they are doing Reimann integration, whereas they're
really doing Lebesgue integration.  From an intuitive physical perspective, there's
not a huge difference.  The way it seems to make a difference is operationally.
They're more likely to USE something in a way consistent with the Lebesgue formalism
without even realizing they're using a theorem from it... it seems like a neat
'trick' to them.

|>
|> Nevertheless, I regularly stick distributions of various kinds under integral
|> signs and come up with correct answers.  I have never studied Lesbegue
|> integration, though I am not proud of that fact.  (You can't study everything -
|> I once got a lecture on Riemann-Steiltjes integrals, which is more
|> than many physicists ever see.)

I've done most of my math studies on my own.  I studied enough calculus on my own to
test out of the sequence by the time I went to college.  I studied Lebesgue
integration by leaving a real analysis book near the toilet (I don't seem to be able
to stand sitting there without something to read -- and I actually found real
analysis fun to read).  While my greatest pleasure is doing physics (music is a close
second), I've often been surprised how few physicists seem to like looking at math
just for the fun of it.  Nevertheless, I find the size of current mathematics to be
absolutely daunting.  One of the basic questions in mathematics is how much of which
theorems will hold under various circumstances (what class of functions can be
Fourier transformed is a typical example), or what kinds of assumptions can lead to
certain results.

While I don't want to have to get into all the details in mathematics, I can say that
there's a few areas in physics that could benefit from the kind of exhaustive
analysis of assumptions that has proven so fruitful for mathematics.  For example,
one of the papers I wrote for Am J Phys lately looked at the Stern Gehrlach
experiment.  I wanted to see how much of it could be explained without resorting to
the measurement/collapse postulate.  Birkhoff's theorem in erogdic theory is based on
the assumption that all of phase space must be sampled by any particular ergodic
trajectory in a long enough time (ever wonder why molecular dynamics simulations or
monte-carlo experiments start showing equilibration and thermal behavior in times so
much shorter than the time for the whole system to sample phase space?).  I think
that looking at the consequences of dropping assumptions, or examining weakened
versions of assumptions can give a much clearer picture of exactly what those
assumptions mean, what their bounds are, how necessary they may be, and what kinds of
alternative ways they can be looked at.

(Sort of long winded for a comment on reading Lebesgue measure.)  Sorry about the
soapbox.


Dan