Absolutely convergent improper integral

2020 Mathematics Subject Classification: Primary: 28A06 [MSN][ZBL]

Definition

Let $f:[a,b[\to \mathbb R$ be a function which is Riemann-integrable on every subinterval $[a,\beta]\subset [a,b[$ (where we also allow the case $b=\infty$). If \begin{equation}\label{e:abs_conv} \lim_{\beta\to b} \int_a^\beta |f(x)|\, dx < \infty\, , \end{equation} then the improper integral \[ \lim_{\beta \to b} \int_a^\beta f(x)\, dx \] exists and is finite and it is said to be absolutely convergent. Similar variants define absolutely convergent improper integrals on $]a,b]$ and $]a,b[$.

Since Riemann-integrable functions are Lebesgue integrable, it actually follows from the definition that $f$ is absolutely integrable and the improper integral of $f$ is just its Lebesgue integral: the notion of absolutely convergent improper integral is therefore seldomly used nowadays.

The existence of the improper integral does not guarantee its absolute convergence, as for instance is the case for the function $\frac{sin x}{x}$ on $]0, \infty$[.

Criteria

Since the absolute convergence of the improper integral of $f$ is guaranteed by the convergence of the improper integral of $|f|$, one can derive classical from those for the [[Improper integral|improper integral. In particular we have

Comparison principle If $g$ is Riemann-integrable, $|g|\leq f$ and the improper integral of $f$ converges, the improper integral of $g$ converges absolutely.

Cauchy criterion The improper integral of $f$ converges if and only if for every $\varepsilon >0$ there exists $\eta\in [a, b[$ such that \[ \int_\eta^\beta |g(x)|\, dx < \varepsilon \qquad \forall \beta\in ]\eta, b[\, . \]

Higher dimensions

The concept of improper integral is difficult to generalize to integrals of several variables, given a certain arbitrariness on how to approximate a general domain. Instead the absence of a canonical approximating sequence of sets is irrelevant for absolutely convergent improper integrals, since one can use the theory of Lebesgue. Indeed, consider any Lebesgue measurable set $\Omega$ and a function $f: \Omega\to \R$ which is Lebesgue integrable on a sequence of measurable sets $\Omega_k\uparrow \Omega$. If \[ \lim_{k\to\infty} \int_{\Omega_k} |f| < \infty \] then $f$ is summable and \[ \lim_{k\to\infty} \int_{\Omega_k} f \] is the Lebesgue integral of $f$ over $\Omega$ (and hence does not depend on the particular sequence $\Omega_k$).

References

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