An old problem: to show that every bounded left integrable function is also Riemann integrable.
I know some (for me) not elementary proofs: Gillespie and Kristensen et al (theorem 1).
The question: can it be an undergraduate problem, i.e. does an elementary proof exist ?
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$\begingroup$ would you give the meaning and the context of left integrable function? $\endgroup$– Pietro MajerApr 15, 2013 at 16:06
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$\begingroup$ @Pietro Majer I refer to Cauchy's definite integral defined by leftsums (1823) $\endgroup$– Antonio PiciulinApr 15, 2013 at 16:25
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1$\begingroup$ Use general finite partitions, as in the Riemann integral, but always evaluate at the left-endpoint of the interval. This is in opposition to alternatives where you use the sup or inf on the intervals, or where you evaluate at an arbitrary point. $\endgroup$– Gerald EdgarApr 15, 2013 at 20:51
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2$\begingroup$ I don't know the answer to your question, but I posted about this result a few years ago in sci.math, and perhaps my comments and additional references could be of use: Math Forum archive of Riemann sums + integrability, sci.math, 7 November 2007. Also, this prior post in the same thread links back to a couple of 2002 posts that might also be of use. $\endgroup$– Dave L RenfroApr 15, 2013 at 21:51
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1 Answer
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The following argument may not be the most direct one, but follows as a quick consequence of the characterization of Riemann integrability, which is perhaps the main result of the theory.
For a bounded function $f:[a,b]\to\mathbb{R}$ and for $\lambda >0$ let's denote $$J _ \lambda :=\{ x\in [a,b]\, : \,\limsup _{y\to x}f(y)-\liminf _{y\to x}f(y) \ge \lambda\}\,.$$ The key point is that for any $\epsilon > 0$ there are two partitions $P^*$ and $P _ *$ of the interval $[a,b]$, both with mesh less than $\epsilon$, such that the corresponding left sums $s_L(f,P^*)$ and $s_L(f,P _ * )$ differ for more than $\lambda\operatorname{meas} (J _ \lambda)-\epsilon$ (to construct $P^*$ and $P _ *$, start from a covering of $J _ \lambda$ by a finite collection of disjoint intervals of length less than $\epsilon$, and whose left endpoints belongs to $J_\lambda$) . As a consequence $$\limsup _ {|P|\to0} s_L(f,P) - \liminf _ {|P|\to0} s_L(f,P) \ge \lambda\operatorname{meas} (J _ \lambda)\, , $$ and we conclude that if $f$ is left integrable in the Cauchy sense, then $\operatorname{meas} (J _ \lambda)=0$ for all $\lambda >0$, that is, it is continuous a.e., hence Riemann integrable by the characterization.
answered Apr 16, 2013 at 7:33
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3$\begingroup$ As to the issue, if it is suitable as an undergraduate problem. The argument above, as well as the characterization of Riemann integrability (which e.g. I usually do in the first year classes of Maths, as "advanced topic"), do not really need the notion of Lebesgue measure, but only the notion of "null set" of the reals (a set that can be covered by a countable collection of intervals with arbitrarily small sum of lengths, and even by a finite collection, in the case of compact sets). So yes, I think it could be given as an elementary (but maybe "challenging") problem for undergraduates. $\endgroup$ Apr 16, 2013 at 8:00