Question

Possible Duplicate:
Is the sum of 2 Lebesgue measurable sets measurable?

Let $A$ and $B$ be two Lebesgue measurable sets in $\mathbb{R}$. Let $C:=\{x+y, x\in A \mbox{ and } y\in B\}$. Is $C$ Lebesgue measurable?

Answer 1

In general, not necessary, even if both $A$, $B$ have measure 0. Take $C=\{\sum_{k=1}^{\infty} \epsilon_k 4^{-k}| \epsilon_k\in\{0,1\}\}$. Call two point equivalent, if corresponding epsilons coincide for large enough $k$. Take exactly one point in each equivalence class and let both $A$ and $B$ be the set of such points. Then $A+B$ has exactly one point in each equivalence class under relation ``to differ by $m/2^n$ for positive integers $m,n$'', and it is standard example of non-measurable set.