Locally Integrable

A locally integrable measurable function on an open set is one for which, around every point in its domain, there is a neighborhood on which it is Lebesgue integrable (Hunter 2011, p. 63). The space of locally integrable functions is denoted . Any Lebesgue integrable function is also locally integrable. One possibility for a nonintegrable function which is locally integrable is if it does not decay at infinity. For instance, is locally integrable on , as is any continuous function on .