If $f:\mathbb{R}\to\mathbb{R}$ is Lebesgue integrable, then $\lim\limits_{t\to 0}\int_\mathbb{R}|f(x+t)-f(x)|dx=0.$

Question

The problem is stated in the title. If $f$ is Riemann integrable, then the Dominated Convergence Theorem and Lebesgue's Criterion for Riemann integrability does it, however I am stuck when the assumption is only Lebesgue integrability.

Answer 1

The continuous functions of compact support $C_c(\mathbb{R})$ are $L^1$ dense in $L^1$. Now play your typical triangle inequality game and you can make this result work.