A neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which $\lim_R\int_{-R}^R|f|dx<\infty$

Question

Is there a neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which the limit of Riemann integrals satisfies $\lim_R\int_{-R}^R|f|dx<\infty$ in terms of elements of $L^1(\mathbb R)$ the space of Lebesgue integrable functions?

Why is this space important for Fourier analysis? I'm guessing this property should be related to being absolutely Riemann integrable, but even if it is, why is that important?

My qusetions are probably elementary but I haven't touched analysis in a while and this class was given a special name in a lecture that was confined to the Riemann integral. I'd like to understand the point without the baggage of Riemann integration...

Also, the lecturer defined $\int_{\infty}^\infty=\lim_R\int _{-R}^R$. This is probably the same question, but why, in the sense of Lebesgue, does everything coincide?

Answer 1

The Riemann integral doesn't make sense for general measurable $f$. Assuming $f$ is Riemann integrable, the Riemann and Lebesgue integrals coincide. Finally, $\lim_{R\rightarrow\infty}\int_{-R}^R|f|dx$ exists iff $f\in L^1$.