7. The Integral

7.5. Riemann versus Lebesgue

In progress ...

If f is L-integrable, so is |f|, but the converse is not true (I think).

There is a function f that is R-integrable but |f| is not R-integrable

Riemann integral for bounded functions only, Lebesgue for bounded and unbounded functions.

Riemann integral can be extended to improper Riemann integrals, but can not allow functions that are extended real valued.

An improper R-integral may exist without the function being L-integrable (see f(x) = sin(x) / x for x > 0.

If f is L-integrable and the improper R-integral exists, then both agree.

Mention sequences of R-integrable function vs. L-integrable

Lebesgue's Theorem: a bounded function f on [a, b] is Riemann integrable if and only if the set of points where f is not continuous has measure zero.

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