[FOM] Category and Measure

[Timothy Y. Chow]

On Tue, 18 Sep 2007, joeshipman at aol.com wrote:
> Such an example would make the best possible case for using the Lebesgue 
> integral instead of the Riemann integral (because it already shows that 
> the Riemann integral fails to be the inverse of differentiability for a 
> *single* function, while the familiar arguments for using Lebesgue 
> rather than Riemann involve its superior properties for *sequences* of 
> functions).

I should also have said that using the Lebesgue integral does not 
automatically eliminate all possible counterexamples to the fundamental 
theorem of calculus.  There's an example in Rudin's "Real and Complex 
Analysis," in the chapter on differentiation, of a function f that is 
continuous on [a,b], that is differentiable at almost every point of 
[a,b], and such that f' is Lebesgue-integrable on [a,b], yet the desired 
formula f(x) - f(a) = int_a^x f'(t) dt fails to hold.  One really needs 
the concept of an absolutely continuous function, in addition to the 
Lebesgue integral, to get the fundamental theorem of calculus.

Rudin also has an exercise in Chapter 2 that I consider to be an excellent 
motivation for the Lebesgue integral.  Let f_n be a sequence of continuous 
functions on [0,1] such that 0 <= f_n <= 1, and suppose that f_n converges 
pointwise to 0.  Show that lim int_0^1 f_n(x) dx = 0.  This is a trivial 
consequence of the dominated convergence theorem, but is annoyingly tricky
to prove without measure theory, even though the statement of the theorem 
requires only the Riemann integral.

While we're on this topic, and to make it more relevant to f.o.m., let
me ask---from a foundational point of view, what other theories of 
integration are there, that arguably have some advantages over the 
Lebesgue theory?

Tim