# continuity and measure

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.

Can you prove it when $A$ and $B$ are both intervals? –  Arturo Magidin Apr 16 '12 at 2:48