This is the inequality we are attempting to prove: Let $f \in \mathcal{L}^p(\mathbb{R}^n), \ g \in \mathcal{L}^1(\mathbb{R}^n)$ and $1 \leq p < \infty$. Then $f * g$ is defined almost everywhere with $f*g \in \mathcal{L}^p(\mathbb{R}^n)$ and $$ \| f * g \|_p \leq \| f \|_p \cdot \|g \|_1. $$
The proof goes as follows:
WLOG we can assume $g \geq 0 $ and $\| g \|_1 = 1$, therefore $ \mu := g\lambda^n $ is a probability measure.
He then goes on to use the Jensen inequality to obtain the result. Why are we able to assume WLOG that $\| g \|_1 = 1$? I'm also not really understanding what $ g\lambda^n$ even means let alone understanding why it would be a probability measure.
Thank you for your help!