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Given a Lebesgue measurable set A with strictly positive measure, can we find an open interval (a,b) such that (a,b) x belongs to A for almost every x in (a,b)?

Thanks in advance for any comments!
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About measurable sets and intervals

Given a Lebesgue measurable set A with strictly positive measure, can we find an open interval (a,b) such that (a,b) belongs to A for almost every x in (a,b)?

Thanks in advance for any comments!