Absolutely integrable function

2020 Mathematics Subject Classification: Primary: 28A20 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

Definition and properties

Consider a measure space $(X, \mathcal{A}, \mu)$. A measurable function $f:X \to [-\infty, \infty]$ is then called absolutely integrable if \[ \int \abs{f}\, d\mu < \infty\, . \] An absolutely integrable function is also commonly called a summable function.

Remark If we assume only the measurability of $|f|$, then this does not guarantee the measurability of $f$. Although a few authors require only the measurability of $|f|$, the vast majority of the literature assumes that $f$ itself is measurable.

The following inequality, which is a particular case of Jensen's inequality, holds for any absolutely integrable function: \[ \abs{\int f\, d\mu}\leq \int \abs{f}\, d\mu \] (the assumption of absolute integrability is however not fundamental: the inequality makes sense and holds as soon as we can define \[ \int f\, d\mu\, , \] that is, as soon as the integral of the positive part of $f$ or that of the negative part of $f$ are finite).

The space of absolutely integrable functions is a linear space which is usually denoted by $L^1 (X, \mu)$ and \[ \|f\|_1 := \int \abs{f}\, d\mu < \infty \] is a seminorm on it. It is customary to identify elements of $L^1 (X, \mu)$ whose values coincide except for a $\mu$-null set: after this identification the norme $\|\cdot\|_1$ endowes $L^1 (X, \mu)$ with a Banach space structure. The $L^1$ space is then just one case of a more general class of Banach spaces called Lp spaces.

Generalizations

The notion of absolutely integrable function can be generalized to mappings taking values in normed vector spaces: in that case $\abs{\cdot}$ is substituted by the corresponding norm. This is straightforward for finite-dimensional vector spaces and all the properties mentioned above holds in this case as well; for the case of infinite-dimensional spaces some care is needed, see Bochner integral.

Lebesgue measure

The primary examples of absolutely integrable functions are given when $X$ is an interval of the real axis (or a domain of $\mathbb{R}^n$), $\mu$ the Lebesgue measure and $\mathcal{A}$ the corresponding $\sigma$-algebra of Lebesgue measurable functions.

Improper integral

Consider an interval $[a,b[$ (resp. $]a,b]$, $]a,b[$) where $b$ might also be $\infty$. Some authors use the term absolutely integrable functions for functions $f$ which are Riemann-integrable on all intervals $[a, \beta]$ with $\beta<b$ (resp. $]\alpha, b]$, $]\alpha, \beta[$) and for which \[ \lim_{\beta\uparrow b} \int_a^\beta \abs{f (x)}\, dx < \infty\, \] (and analogous conditions for the other cases). This implies the existence (and finiteness) of \[ \lim_{\beta\uparrow b} \int_a^\beta f (x)\, dx\, \] (and analogous limits for the other cases), which is often called Improper integral.

The converse is not true, namely the existence of the improper integral does not guarantee the absolute integrability, regardless whether we are dealing with Riemann-integrable or Lebesgue-integrable functions. An example is \[ \int_0^\infty \frac{\sin x}{x}\, dx\, . \]

References

[AB]	C.D. Aliprantz, O. Burleinshaw, "Principles of real analysis" , North-Holland (1981)
[IP]	V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[Ku]	L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) MR1617334 MR1070567 MR1070566 MR1070565 MR0866891 MR0767983 MR0767982 MR0628614 MR0619214 Zbl 1080.00002 Zbl 1080.00001 Zbl 1060.26002 Zbl 0869.00003 Zbl 0696.26002 Zbl 0703.26001 Zbl 0609.00001 Zbl 0632.26001 Zbl 0485.26002 Zbl 0485.26001
[Nik]	S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian) Zbl 0397.00003 Zbl 0384.00004
[Roy]	H.L. Royden, "Real analysis" , Macmillan (1968) MR1013117 MR1532990 MR0151555 Zbl 1191.26002 Zbl 0704.26006 Zbl 0197.03501 Zbl 0121.05501
[Ru1]	W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) MR0055409 Zbl 0052.05301
[Ru2]	W. Rudin, "Real and complex analysis" , McGraw-Hill (1966) pp. 98 MR0210528 Zbl 0142.01701
[Sch]	L. Schwartz, "Cours d'analyse" , 1 , Hermann (1967) MR0756815 MR0756814
[Tay]	A.E. Taylor, "General theory of functions and integration" , Blaisdell (1965) MR0178100 Zbl 0135.11301
[Zaa]	A.C. Zaanen, "Integration" , North-Holland (1967) MR0222234 Zbl 0175.05002