Young's Inequality for Convolutions/Corollary 2

Corollary to Young's Inequality for Functions
Let $f, g: \R^n \to \R$ be Lebesgue integrable functions.

Then their convolution $f * g$ is also Lebesgue integrable. Thus, convolution may be seen as a binary operation $*: \mathcal{L}^1 \times \mathcal{L}^1 \to \mathcal{L}^1$ on the space of integrable functions $\mathcal{L}^1$.

Proof
Apply Young's Inequality for Functions with $p = 1$.