In a failed attempt to solve this seemingly straightforward problem...
How do I show that a distribution is locally p-integrable?
...I realized that I don't understand these spaces as deeply as I would like to. For example, my (naive) way of thinking of a function $f \in L^p_{loc}$ is as a "pseudo"-$L^p$ function which is only integrable on "small" (compact) regions. Is that the right intuition? If not, what would be a better way of thinking about it?
A few more questions to address
1) Other than their definitions, what are some major conceptual and practical differences between the $L^p$ and $L^p_{loc}$ spaces?
2) Which non-trivial classes (if any) are dense in $L^p_{loc}$
3) Which are the most commonly used methods and techniques when dealing with the above spaces? (Ex: Holder's inequality? Density arguments? etc)