Regular distribution function

Question

For a regular distribution F we know it can be identified by a locally integrable function s.t $$F(\phi) = \int_{ℝ^N}f(x)\phi(x)dx \space,\space x \in ℝ^N \space,\space \phi(x) \in D(ℝ^N) $$ I've assumed the function f is unique for a given distribution but I can't find anything to confirm or explain why this is, I know this is kind of a simple question but I'm not getting any help from my book or google.

Answer 1

$f$ will be unique up to sets of measure zero. Remember, $f$ really isn't a function but rather an equivalence class of functions who are equal a.e. i.e. a member of $L^1_{loc}$. So if we define the distribution as the integration of a smooth function of compact support again a member of $L^1_{loc}$ then the uniqueness is really a result of integration. Try working it out, suppose you have two different $f_1,f_2 \in L^1_{loc}$ what can you say about the distributions they define in the above manner?