Why we need $\sigma$ finite measure to define $L^{p}$ space? I am curious why we need $\sigma$ finite measure to define $L^{p}$ space. More generally, Why we need $\sigma$ finite measure instead of just finite measure? 

You can define $L^p$ on any measure space you like.  If it just those measurable functions for which $|f|^p$ is integrable.  And then you modulo by functions zero a.e.
But if you don't impose some hypothesis like $\sigma$ finite, many results about $L^p(\mathbb R)$ don't always generalize very well to arbitrary measure spaces.
