Re: The logical structure of calculus, request for help.

Herman Rubin wrote:
It is easier to learn measure and integration, and the
differences between Riemann-type and Lebesgue-type
completions of integrals of functions with finitely
may values on "appropriate" sets.

Riemann integration will work just as well on functions
defined on the rationals only; the extension can be made using finitely additive measures and integrals,
such as limits. Lebesgue type extensions can be easily
made using countable approximations, and the measure
theory needed done quickly without the usual cute means
of introducing measurability.

But do not start with length measure on the line; start
with discrete measures on finite and countable spaces,
and pass to limits.  This is all done easily and keeps
both the rigor and the intuition.


Some of your wording is foreign to me. Are you agreeing with Spriggs and then showing the pros and cons of both types of integrals?

I've taken the required calculus courses for my physics degree and have not been taught anything that I know of about the Lebesgue integral. Might there be a reason for that?

However, I've also been taught Hamiltonian dynamics without having first taken a course in the calculus of variations, so it is quite possible that I've something of the Lebesgue integral without knowing it.

Thanks, Adam.
.

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