I've always been bugged by examples of functions that are (improperly) Riemann integrable, but not Lebesgue integrable, like $\frac{\sin x}{x}$. While I understand perfectly well why such a function is not Lebesgue integrable, I can't help but wonder if there is some meaningful way to extend the idea of the Lebesgue integral, perhaps as an "improper" Lebesgue integral.
The first idea that comes to mind is looking at all measurable functions which have bounded $L^p$ norms over all bounded intervals:
$$\mathcal{L}^p(\mathbb{R}) := \left\{\, f : \sup_{-\infty < a < b < \infty} \left|\left| \,\chi_{(a,b)} \,f \,\right|\right|_{L^P} < \infty \right\},$$
where $\chi_{(a,b)}$ is the characteristic function of the interval $(a,b)$.
I realize that we are being a bit arbitrary in the requirement that the intervals be bounded, but at the same time we aren't restricting ourselves to something as rigid as symmetric intervals (like a principal value integral might do). Would such a space have any meaningful properties, or would all of the important theorems from measure theory and functional analysis fall apart? And is there any nice way of classifying the functions that live in the space $\mathcal{L}^p(\mathbb{R})\setminus L^p(\mathbb{R})$?
