NetNews Usenet Archive 1992 #20/NN_1992_20.iso/spool/sci/math/11183

Internet Message Format | 1992-09-09 | 2.4 KB

Path: sparky!uunet!europa.asd.contel.com!darwin.sura.net!wupost!waikato.ac.nz!comp.vuw.ac.nz!cc-server4.massey.ac.nz![e-mail]
Newsgroups: sci.math
Subject: Re: Couple of questions
Message-ID: <[e-mail]>
From: [e-mail] (USENET News System)
Date: Thu, 10 Sep 92 00:31:12 GMT
References: <[e-mail]> <[e-mail]>
Organization: Massey University
Lines: 41

In article <[e-mail]>, [e-mail] (John C. Baez) writes:
>
> Very roughly, the way Lebesgue integration works is to slice the graph of
> the function *horizontally* rather than *vertically*.  But this only
> half of the story - the easy half.  When you do the slicing, you can
> get some rather nasty-shaped slices if you're dealing with a nasty
> discontinuous function.  To figure out how long these slices are you
> need to develop the notion of Lebesgue measure.

Titchmarsh "The Theory of Functions" says
"Perhaps the most obvious difference [between Riemann and Lebesgue
integration] to the beginner [not that I'm saying you are :-)] is that,
in Lebesgue's definition, we divide up the interval of variation of the
function instead of the interval of integration. This, however, is
comparatively unimportant. What is essential is that we use the general
theory of 'measure' of sets instead of the more limited theory of
'extent'. It would be possible to build up an integral from integrals of
characteristic functions, but using extent instead of measure. This
would be substantially equivalent to Riemann's definition. On the
other hand, it is possible to define an integral equivalent to Lebesgue's
by dividing up the interval of integration in a suitable way."

It is a common fallacy that the function is divided into horizontal
strips, actually the strips are still vertical but determined by the
values of x such that a < f(x) < b; and they are not rectangular but
product sets.

I'm sure you know this - these myths creep in when one tries to
simplify the definition for the masses.

When I try to motivate the Lebesgue integral I use an example like
f(x) = x if x is rational with denominator a power of 2,
f(x) = 0 if x is rational with denominator not a power of 2,
f(x) = x^2 otherwise.
Then say something like "we want to see how much of the interval
of integration gives y < f(x) < y + dy".

Terry Moore