Multiplication of Distribution induced by Locally Integrable Function by Smooth Function

Theorem
Let $f \in \map {L^1_{loc} } {\R^d}$ be a locally integrable function.

Let $\alpha \in \map {\CC^\infty} {\R^d}$ be a smooth function.

Let $T_f \in \map {\DD'} {\R^d}$ be a distribution induced by $f$.

Then in the distributional sense it holds that:


 * $\alpha T_f = T_{\alpha f}$

Proof
Let $\Omega \subseteq \R^d$ be a compact subset.

Then for all $\mathbf x \in \Omega$ we have that $\map \alpha {\mathbf x}$ is bounded.

Hence, $f \alpha$ is locally integrable.

Let $\phi \in \map \DD {\R^d}$ be a test function.

Then: