[FOM] Category and Measure
Timothy Y. Chow
tchow at alum.mit.edu
Tue Sep 18 18:05:11 EDT 2007
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
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