Equivalence of Almost Equal Locally Integrable Functions and their Distributions

Theorem
Let $f, g \in \map {L^1_{loc}} \R$ be locally integrable functions.

Let $T_f, T_g$ be the distributions associated with $f$ and $g$ respectively.

Then the following statements are equivalent:


 * $T_f = T_g$


 * For almost all $\mathbf x \in \R^d$ we have $\map f {\mathbf x} = \map g {\mathbf x}$.