I have somehow confused myself with this fairly straightforward proof. We need to show that $\lambda f$ is a measurable function on $(S,\mathcal{S})$, i.e. that for any $c\in \mathbb{R}:\{s\in S:f(s)\le c\}\subset S$. If $\lambda \ne 0$, then the claim follows immediately from the measurability of $f$, namely as $c/\lambda \in \mathbb{R}$ it follows that $\{s\in S:f(s)\le c/\lambda \}=\{s\in S:\lambda f(s)\le c\}$

But then, if $\lambda =0,\lambda f=0$ and I am not sure how to convince myself that $\lambda f$ is measurable.

But then, if $\lambda =0,\lambda f=0$ and I am not sure how to convince myself that $\lambda f$ is measurable.