Convolution Inequality. Let $f \in \mathcal{L}^p(\lambda^d), \ g \in \mathcal{L}^1(\lambda^d)$ and $1 \leq p < \infty$. Then $f * g$ is defined almost everywhere with $f*g \in \mathcal{L}^p(\lambda^d)$ and $$ \| f * g \|_p \leq \| f \|_p \cdot \|g \|_1. $$
Step 1 in the proof of my professor: By linearity wrt to $g = g^+ - g^-$ we can let $g \geq 0$ and WLOG $\|g \|_1 = 1$. (So we can apply Jensen later.)
I seem to be overlooking something but how does linearity (of $*$ I assume) permit us to let $g \geq 0$?
Because if you have established Young's inequality for $g \ge 0$ then for arbitrary $g$ you have $$f \ast g = f \ast (g^+ - g^-) = f\ast g^+ - f \ast g^-$$ so that $f \ast g$ is defined almost everywhere and $$ \|f \ast g\|_p = \|f\ast g^+ - f \ast g^-\|_p \le \| f\ast g^+ \|_p + \|f \ast g^-\|_p \le \|f|_p \|g^+\|_1 + \|f\|_p \|g^-\|_1$$ where $$\|g^+\|_1 + \|g^-\|_1 = \int g^+ + \int g^- = \int_{\{g \ge 0\}} |g| + \int_{\{g < 0\}} |g| = \|g\|_1.$$