Which integrable functions fail to be absolutely integrable? (English) Zbl 1400.26003

Mendoza, José

Real Anal. Exch. 43, No. 1, 243-248 (2018).

In the present paper, the author considers a natural question: if we consider real-valued functions defined on an interval \([a,b]\), is it possible to construct examples of Riemann integrable but not absolutely integrable functions? Independently of that one thinks that the answer should be positive, the author establishes that in some way, the answer is “no”. More precisely, the following result is proved: assume that \(f:[a,b]\to \mathbb{R}\) is an integrable but not absolutely integrable function. Then there exists a strictly monotone sequence \(\{{{t}_{n}}\}\subset [a,b]\) such that the series \(\sum\limits_{n=1}^{\infty }{\int\limits_{{{t}_{n}}}^{{{t}_{n+1}}}{f(x)dx}}\) is convergent but not absolutely convergent.

Reviewer: Andrey Zahariev (Plovdiv)

MSC:

26A03	Foundations: limits and generalizations, elementary topology of the line
26A99	Functions of one variable

Keywords:

integrable function; absolutely integrable function; conditionally convergent series