I'm quite sure there is a name for such class of functions but can not remember or figure out what terms to search for. A simple and very practical example would be "periodic Lebesgue" functions: functions which are periodic—and thus, except for null cases, can not in principle have a Lebesgue integral over $\mathbb{R}$–but are integrable in their restriction to the fundamental interval. Or any finite interval whatsoever, for that matter.
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1Locally integrable. – Oct 23 '13 at 13:14
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@T.Bongers: Exactly what I was looking for, thanks for the push! If it's worth an answer to you, I could accept it. Otherwise, I guess I'll delete the question for being too trivial. – The Vee Oct 23 '13 at 13:17
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Such a function is called locally integrable, and the space of such functions is generally notated as $L^1_{\text{loc}}$ for this reason. An alternative, and equivalent, definition, is that for all infinitely differentiable $\varphi$ with compact support, we have
$$\int |f \varphi| dx < \infty$$