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Let $ A, B \subseteq \mathbb {R} $ be Lebesgue measurable sets such that at least one of them has finite measure. Let $ f $ be the function defined by $$f (x) = m ((x + A) \cap B)$$ for each $ x \in \mathbb{R} $. Show that $ f $ is continuous.

Hint: Suppose first that $ A $ and $ B $ are intervals and then generalized to arbitrary sets using the regularity of the Lebesgue measure.

Some help please. Thanks.

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    Can you prove it when $A$ and $B$ are both intervals?2012-04-16

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