Continuity of the lebesgue integral How does one show that the function, $g(t) = \int \chi_{A+t} f $ is continuous, given that $A$ is measurable, $f$ is integrable and $A+t = \{x+t: x \in A\}$.
Any help would be appreciated, thanks

Notice that 
$$
|g(t+h)-g(t)| \le \int_{(A+t)\Delta A} |f|
$$
so it is enough to prove that
$$
  |(A+t)\Delta A| \to 0 \qquad \text{as }t \to 0
$$
where $\Delta$ is the symmetric difference, since 
$$
\int_{A_k} f \to 0 
$$
if $|A_k|\to 0$.
