Let $B$ be a bounded Borel set of $\mathbb{R}$, Show that if $A$ is a finite union of disjoint intervals, the Lebesgue measure of $A\triangle B$ can be arbitrarily small. Also show that this remains true as long as $B$ has finite Lebesgue measure.
Approximating Borel sets by finite unions of intervals
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measure-theory
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1Sometimes this is taken to be the definition of a measurable set. – 2012-10-19