# Cauchy's left endpoint integral (1823)

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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would you give the meaning and the context of left integrable function? – Pietro Majer Apr 15 '13 at 16:06
@Pietro Majer I refer to Cauchy's definite integral defined by leftsums (1823) – Antonio Piciulin Apr 15 '13 at 16:25
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. – Gerald Edgar Apr 15 '13 at 20:51
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. – Dave L Renfro Apr 15 '13 at 21:51

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.