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 $\map f {\mathbf x} = \map g {\mathbf x}$.

Proof

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Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples