Is the sum of Lebesgue-zero sets still a Lebesgue-zero set?

Asked Mar 14 '21 at 19:47

Active Mar 14 '21 at 21:20

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Question: Given $A,B \subseteq \mathbb{R}$ with lebesgue-measure zero, is the set $$ A + B := \{ a + b \mid a \in A,\,b \in B\}$$ also of lebesgue-measure zero?

I have not yet taken any measure theory courses; up until now I know the Lebesgue integral and what a measure zero set is. I cannot think of an example (or counter-example) myself, and am keen to see what you think.

edited Mar 14 '21 at 21:20

Asaf Karagila

405,794

asked Mar 14 '21 at 19:47

Philippe Knecht

Yes for a countable sum! But not if you take an uncountable sum, and you should see some easy counter examples there :) – Osama Ghani Mar 14 '21 at 19:50
6

Not necessarily: if $C$ is the middle-thirds Cantor set, $C+C=[0,2]$. See here, the links in Martin Sleziak’s comment under that question, and the link in my answer to that question. – Brian M. Scott Mar 14 '21 at 19:52
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@OsamaGhani See Brian's comment! (Possible point of of confusion here: "sum" is not "union" in this context.) – Noah Schweber Mar 14 '21 at 20:33
Oops, definitely misunderstood! – Osama Ghani Mar 15 '21 at 00:07

Is the sum of Lebesgue-zero sets still a Lebesgue-zero set?

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