Can someone give an example of a function that is not Henstock-Kurzweil/gauge integrable?

Question

I am looking for an example of a function that is not Henstock-Kurzweil integrable. Can anybody help me?

Answer 1

Take any infinite series $a$, and let $f(x)$ be $a_{\lfloor{x}\rfloor}$ (where $\lfloor{x}\rfloor$ is the greatest integer that is not greater than $x$, that is $x$ rounded down to an integer). In general, $$ \int_i^j f(x) \,\mathrm{d}x = \sum_{n = i}^{j-1} a_n ,$$ whenever $i$ and $j$ are integers. As stated, this is a fine Riemann integral, but change $j$ to $\infty$ and we have:

• $ \int_i^\infty f(x) \,\mathrm{d}x = \sum_{n = i}^\infty a_n $ as a Lebesgue integral iff $a$ is absolutely convergent;
• $ \int_i^\infty f(x) \,\mathrm{d}x = \sum_{n = i}^\infty a_n $ as an improper Riemann integral iff $a$ is convergent (possibly conditionally);
• $ \int_i^\infty f(x) \,\mathrm{d}x = \sum_{n = i}^\infty a_n $ also as a Henstock–Kurzweil integral iff $a$ is convergent (possibly conditionally).

So any divergent series gives a Henstock–Kurzweil nonintegrable function. The harmonic series, for example, gives $f(x) = 1/\lfloor{x}\rfloor$, which is not integrable.

OK, so just plain $1/x$ is an even simpler nonintegrable function, but I think that if you keep in mind the relationship between Lebesgue vs Henstock and absolute vs conditional convergence, then it will seem more obvious to you that, while Henstock–Kurzweil makes more functions integrable, it does not make all functions integrable.

Answer 2

Theorem. If function $f: [a,b] \to \mathbb{R}$ is Henstock-Kurzweil integrable, then $f$ is a Lebesgue measurable function.

You can find proof of the above theorem in "The integrals of Lebesgue, Denjoy, Perron, and Henstock" by Russel Gordon (Theorem 9.12).

Thus a characteristic function of a non-measurable set would be an example you are looking for.