Absolutely integrable function
2020 Mathematics Subject Classification: Primary: 28A20 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$
Contents
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
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Absolutely integrable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely_integrable_function&oldid=29892