How do you call functions integrable over any compact subset of their domain?

Question

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.

Answer 1

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

$$\int |f \varphi| dx < \infty$$