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

Assuming the integral is using Lebesgue measure over ${\Bbb R}$, one approach is to use the fact that continuous functions with compact support are dense in $L^1$.  Then, observe that if $f$ is continuous with compact support, $x\mapsto f(x+h)$ converges uniformly to $f$ as $h\to 0$.
