I have to prove Brunn-Minkowski inequality:
$K$, $L$ are compact sets in $\mathbb{R}^n$, I want to prove that $|K+L|^{\tfrac{1}{n}} \ge |K|^{\tfrac{1}{n}} + |L|^{\tfrac{1}{n}}$, where $X+Y$ denotes Minkowski sum, and $| \cdot |$ is Lebesgue measure on $\mathbb{R}^n$.
My attempts were as follows. I proved the inequality for $n$-dimensional boxes then I generalized the inequality for a finite sum of boxes.
Now I'm wondering how to prove inequality for open sets. I know that every open set in $\mathbb{R}^n$ can be approximated by a finite sum of boxes, so I can take such approximations $K_i$, $L_i$ and write $|K_i+L_i|^{\tfrac{1}{n}} \ge |K_i|^{\tfrac{1}{n}} + |L_i|^{\tfrac{1}{n}}$, but there is a problem. It is known that $\lim_{i \to \infty} |L_i| = |L|$, but what about $\lim_{i \to \infty} |K_i+L_i|$? Probably it is obvious that $\lim_{i \to \infty} |K_i+L_i|=|K+L|$, but I don't know why. It seems that the continuity of measure is not enough, I need to know something about Minkowski's sum.
If I prove inequality for open sets, the rest will be easy.