# Characterizing Riemann-integrability via condition on indefinite Lebesgue-integral

Let $f$: $I\longrightarrow\mathbb{R}$ be Lebesgue-integrable and let $F$ be some indefinite integral of $f$. We know that $F$ is absolutely continuous and hence differentiable a.e. with $F'=f$. In fact, this is a characterizing property, so $f$ is Lebesgue-integrable if and only if it is expressible as the a.e. derivative of an absolutely continuous function.

Now suppose that in the last statement we replace Lebesgue- by Riemann-.

How can we modify the condition on the right so as to maintain equivalence?

Simpler question: What relevant additional properties does $F$ have when $f$ is Riemann-integrable?

• I don't know if it can help, but read this paper by Berberian, 5, p. 210: the class of "primitives" $\dots$ – Tony Piccolo Jan 2 '16 at 16:32

I remember at a summer conference Erik Talvila asked me this question, as he had asked some others:

Erik's Problem. What conditions on a function $F:[a,b]\to\mathbb R$ are necessary and sufficient in order that $$F(x)= \int_a^x f(t)\,dt + C$$ for some constant $C$ and some Riemann integrable function $f$?

He posed it more formally in

E. Talvila, Characterizing integrals of Riemann integrable functions, Real Anal. Exchange, 33(2) (2007), 487–488.

I flippantly said, at the time, that $F$ was Lipschitz with a derivative that is equivalent to an a.e. continuous function. But we both knew that was a silly (if correct) answer. Maybe a slightly less silly answer is that $F$ is Lipschitz with a strong derivative at almost every point. [The strong or strict or unstraddled derivative has been discussed on this site several times.] The reason that works is that the existence of the strong derivative at a point requires the ordinary derivative to exist a.e. in a neighborhood of that point and be continuous at the point.

Compare it with these questions:

PROBLEM A. What conditions on a function $F:[a,b]\to\mathbb R$ are necessary and sufficient in order that $$F(x)= \int_a^x f(t)\,dt + C$$ for some constant $C$ and some Lebesgue integrable function $f$?

[Answer: $F$ is absolutely continuous.]

PROBLEM B. What conditions on a function $F:[a,b]\to\mathbb R$ are necessary and sufficient in order that $$F(x)= \int_a^x f(t)\,dt + C$$ for some constant $C$ and some Denjoy-Perron integrable function $f$?

[Answer: $F$ is ACG${}_*$ (i.e., generalized absolutely continuous).]

PROBLEM C. What conditions on a function $F:[a,b]\to\mathbb R$ are necessary and sufficient in order that $$F(x)= \int_a^x f(t)\,dt + C$$ for some constant $C$ and some function $f$ of bounded variation?

[Answer: See F. Riesz, Sur certain syst`emes singulier d’equations int´egrales, Annales de L’Ecole Norm. Sup., Paris (3) 28 (1911), 33–62.]

Problems A, B, and C have quite satisfying answers as you can see. In the spirit of Riesz's characterization for the latter problem (and embarrassed that I couldn't find a fast answer) I published this as an answer to Erik's problem:

Thomson, Brian S. Characterizations of an indefinite Riemann integral. Real Anal. Exchange 35 (2010), no. 2, 487--492.

I doubt you will find the characterization all that compelling. Don't be discouraged from finding a better one!

• Many thanks for this! Could it be something along the lines of "absolutely continuous + extreme derivatives in Baire class 1"? – Damian Reding Jan 2 '16 at 18:44
• Not quite. There is a Lipschitz function $F$ that is everywhere differentiable but $F'$ is not Riemann integrable (although it is Lebesgue integrable). The function $F'$ itself is Baire 1 of course. – B. S. Thomson Jan 2 '16 at 19:01
• I actually do find it compelling, because, given this characterization, the condition (3) in your paper leads one to consider (via the "differentiable implies continuous" intuition) the definition of what we know is absolute continuity and hence, by analogy, to predict the existence of a more general integral. I try to think of this as a natural way to build up the class of Lebesgue-integrable functions from the class of Riemann-integrable ones. – Damian Reding Jan 3 '16 at 0:50
• Interesting. One wonders whether this problem, had it been posed in the 19th century, would have had a satisfactory solution and whether that might have led to a more general theory of integration. – B. S. Thomson Jan 3 '16 at 1:09