From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Re: Couple of questions
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Wed, 9 Sep 92 17:49:10 GMT

It's a tall order to ask for an explanation of Riemann and Lebesgue
integration, but I'll give it a shot.  It'll necessarily be sketchy,
since Lebesgue integration, in particular, is rather sophisticated,
and typically takes a whole course to master.  But I'll try to give
you the idea.

In Riemann integration you calculate the integral of a function by
figuring out the area under its graph.  (Let's assume for simplicity
that the function is nonnegative!)  Fine, but how do you figure out
the area under the graph.  Well, you chop the graph into lots of
vertical strips, approximate each strip by a rectangle, and add up the
areas of the rectangles.  This gives an approximate answer.  Then you
take the limit as your strips get skinnier and skinnier - if the limit
exists, we say the Riemann integral exists, and equals that limit.  A
basic theorem is that the Riemann integral of any continuous function
on a closed interval [a,b] exists.  Some discontinuous functions are
Riemann integrable too.  For example, the function on [0,2] that
equals 0 on [0,1) and 1 on [1,2] - or any such "step function".  

There are, however, lots of functions that are not Riemann integrable. 
In fact, in some sense "most" functions are not Riemann integrable.  A
typical one is the function on [0,1] that equals 0 on all the rational
numbers and 1 on all the irrational numbers.  This is just too
discontinuous to be Riemann integrable - it's discontinuous at every
single point, in fact.  The irritating thing is that, after some
thought, it's obvious what its integral "should be": 1.  Why?  Well,
there are only countably many rational numbers, but uncountably many
irrationals.  So the set on which this function equals 0 should be
"negligable" in some sense, and the integral should be just the same
as the integral from 0 to 1 of the constant function 1 -- namely, 1!

Lebesgue integration is a way to make this precise.  In particular,
when you learn Lebesgue integration you learn the technical sense in
which the above function is equal to 1 "almost everywhere", and equal
to 0 only a set of "measure zero".  

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.  For example, we say
that the set of irrational numbers between 0 and 1 has Lebesgue
measure 1, while the set of rationals has Lebesgure measure 0.

I hope this whets your curiosity to learn Lebesgue integration, and
gives you some of the intuition about it that you need in order to
work your way through the rather technical details.  The first math
course I really loved was the course on real analysis in which I
learned Lebesgue integration.  It was the teacher (Robin Graham) who
did it, as much as the material.  As a result I became something of an
analyst - until recently, when forced to pigeonhole myself as a
mathematician, I would say "functional analyst" - though I usually
thought of myself as a mathematical physicist, which means practically
whatever you want it to mean, as long as you know F = ma.  :-)


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