Re: Riemann Integral and discontinuities

In article <hg1Uf.5457$ji6.304880@xxxxxxxxxxxxxxxxxxxxx>,
True Raptor <CB4ever@xxxxxxxxxxxxxxx> wrote:

Concerning the Riemann Integral, is it possible that it can properly
integrate a function over certain types of discontinuities, WITHOUT
having to resort to improper integral techniques (approaching the
discontinuities with limits). And what types of discontinuities can
this be done over?

Many have posted the answer that the necessary and
sufficient condition that the set of discontinuities has
Lebesgue measure 0. But there is an equivalent condition
which does not use even the idea of sets of measure 0; the
set of discontinuities of size greater than c, for any c>0,
is contained in a union of a finite number of intervals
whose total size is arbitrarily small. Proving this one
is easy.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

Relevant Pages

  • Re: Simple questions about topology
    ... the indicator of the cantor ternary set is not Riemman integrable as the set of discontinuities is not countable. ... are those of the Statistics Department or of Purdue University. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)
  • Re: RV and CLT
    ... are those of the Statistics Department or of Purdue University. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)
  • Re: Are L^p spaces separable?
    ... A necessary and sufficient condition that L^p spaces are ... separable (0 < p < infinity) is that there is a countable ... are those of the Statistics Department or of Purdue University. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)