Definition:Integrable Function/Unbounded Above

Definition

Let $f: \R \to \R$ be a positive real function.

Let $f$ be unbounded above on the open interval $\openint a b$.

Let $f_m$ denote the function $f$ truncated by $m$ for $m \in \Z_{>0}$.

Suppose that $f_m$ is Darboux integrable on $\openint a b$ for all $m \in \Z_{>0}$.

Suppose also that the following limit exists:

$\ds \lim_{m \mathop \to \infty} \int_a^b \map {f_m} x \rd x$

Then $f$ is integrable on $\openint a b$ and can be expressed in the conventional notation of the Darboux integral:

$\ds \int_a^b \map f x \rd x$

Sources

• 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions