Let $S, T \subseteq \mathbb{R}^n$ be measurable, and let $S-T = \{x-y: x\in S, y\in T\}$.
I know that it has been showed that $S-T$ is not necessarily measurable, but can we say anything about a set $W$ containing $S-T$ and which has measure somehow bounded by the measures of $S$ and $T$? Perhaps it's a stretch but it would be nice.
I think Alphonse's link gives a good answer, so I'm just going to stick that here so the question reads as "answered."
The tl;dr of that link is that you might have two sets of measure 0 whose sum (or difference, in this case) has large measure. We can take $S = [0, 1] \times \{0\}$ and $T = \{0\} \times [-1, 0]$ in $\mathbb{R}^{2}$. Both have measure 0, but $S - T = [0, 1]^{2}$, which has measure 1.