Definition:Integrable Function/Unbounded Above

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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:

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


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

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


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