Why is the metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ important? The metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ appeared twice when I was studying. The author said that the space of Riemann integrable function with the metric $d$ is not complete, but the space $L^1$ yes, it is.
But... There is metrics that makes a lot of function spaces complete. Why is the metric $d$ so special? It has a name?
(apologize the bad English, please)
 A: The map $f \mapsto \int_a^b f(x)\,dx$ is continuous with respect to the metric $d$.  So if you have a sequence $f_n \to f$ converging in the metric $d$, you know that $\int_a^b f_n(x)\,dx \to \int_a^b f(x)\,dx$.  In other words, convergence in the metric $d$ is exactly the right thing to guarantee that you can pass the limit under the integral sign.
A: In general in analysis it is desirable to have objects of interest such as functions classified in spaces with "good properties". This allows one to use some standardised proof techniques for some problems and is very convenient. Completeness turns out to be one of the most important of such properties. Also, integration is one of the fundamental operations in real analysis. One can show that integrable functions which are obviously of great interest form a very nice space when equipped with this distance function, which comes from so called $L^1$ norm. It's one of whole family of $L^p$ norms (all defined by integrals, you can look it up). $L^p$ spaces with these norms turn out to be complete, therefore Banach spaces for all $p \in [1,\infty)$. In case $p=2$ it's even Hilbert. On top of that there is Holder's inequality. 
To sum it up, importance of this metric relies on its good properties. If you continue studying analysis you will see spaces I described over and over again. Also this metric is quite natural measure of distance of functions - it simply measures the area under the graph of difference. 
