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Notation: $ A+B = \{ a + b : a \in A, b \in B \}. $ H. Steinhaus proved the classical result that $ A+B $ contains an interval if $ A $ and $ B $ are both measurable subsets of the real line, each having positive Lebesgue measure. Can we have measurable sets where one set has positive measure and the other has zero measure but their addition contains an interval? I have tried to find an example which disproves this but I couldn't. Thank you for help.

user1008646
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Uswadkar Prashant Vasantrao
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    As of what the question is asking right now, I'd say $A=(0,1)$ and $B={x}$ any singleton. $A+B=(x,1+x)$ –  Jun 21 '16 at 06:04

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Let $C$ be the standard Cantor set. $C$ has measure zero and $C+C=[0,2];$ see this question.

Let $S$ be any set of positive measure. Let $A=C$ and let $B=C\cup S;$ then $A$ has measure zero, $B$ has positive measure, and $A+B\supseteq C+C=[0,2].$

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bof
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Remark: I thought the OP was looking for an example where the addition does not contain an interval, so this is not an answer to the original question.

Take the fat Cantor set for $A$ and a point, for example $2$, for $B$ - or even the empty set, to make it more trivial.

You then get a disjoint union of copies of the fat Cantor set, each of which does not contain any interval

b00n heT
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