archiver 8

Sunday, February 19, 2006

condition for Riemann integrability

criteria for being Riemann integrable
tag: math. proof. divide. sci.math. integrable. criteria. using compact. using refinement. using covering. oscillation. envelope. using closure. using point set topology. proving measure zero. history.

f : [0,1] or [a,b] --> [-M,M] or [-1,1]
D = the set of points where f is discontinuous
D_e = the set of points where f is discontinuous by size larger than e somehow.

"f (bounded) is Riemann integrable on [0,1] iff m(D) = 0"

It says f is Riemann integrable iff the set of badly-behaving points D is small.

See Robert Israel's nice summary of a proof (the 'if' part) in which he directly successfully uses D rather than using D_e in constructing the required partition of [0,1]. Rather than to use directly D and K = [0,1]\D, he covers (and replace) D with a slightly bigger open set U and K with V. (compactness of K is used to construct V and finiteness of F1 is used to construct F2). Uses F1 and F2 to find a refinement partition pi. pi is not directly constructed from D and K, but by first covering and replacing D and K both by F1 and F2 (finite collection of open intervals).
learn: flexibility. using refinments. using covering.
concept: covering. refinement. compact.

See Herman Rubin's proof (of both 'if' and 'only if' part at once) in which he uses D_e (in his notation, A_e) rather than using directly D. He uses that 'the closure of A_e is contained in A_(e/2)'. this is for using compactness to reduce to 'finitely many' intervals.
learn: using closure containment.
concept: closure containement. compact.


off topic.

See Robert Israel's 'only if' proof. (a usual and straightforward proof). The proof can be considered 'divide and conquer' twice. He reduces
m(D)=0
to
m(D_e) < (arbitrary small).
First 'divide and conquere', in that D --> D_e.
Second 'divide and conquere', in that 0 --> arbitrary small.
Observe that this is usual in proving something is measure zero.
learn: "take epsilon -> 0". usual way of proving m(something)=0.

See R3769's where the concept of lower envelope and upper envelope is used.

See Dave L. Renfro's post, for the history. an excerpt:
"Riemann's nontrivial contributions on this topic were (a) giving a
necessary and sufficient condition for integrability based on the
behavior of a function, (b) using this condition to prove the
integrability of a function having a dense set of discontinuities,
and (c) putting the focus on the collection of functions that are
integrable according to some notion of integrability, rather than
defining a notion of integrability only in order to rigorously
prove certain desired integrability properties."

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