## Lebesgue measurability of the Minkowski sum of Lebesgue measurable sets [closed]

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?

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 This exact question has been asked and answered before. The answer is no. mathoverflow.net/questions/19471/…. – Pablo Shmerkin Nov 26 2010 at 21:00

## closed as exact duplicate by Qiaochu Yuan, Andrey Rekalo, Andres Caicedo, François G. Dorais♦Nov 26 2010 at 21:17

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.