I'd really appreciate your help with the following problem:
Let $f$ be a integrable function in $[a-1,b+1]$. I need to prove that: $\lim_{h\to 0} \int_{a}^{b} |f(x+h)-f(x)|dx=0 .$
Basically I need to show that $\forall \varepsilon \exists \delta .|h|< \delta \to \left|\int_{a}^{b}|f(x+h)-f(x)|dx\right| < \varepsilon $.
In addition, I know that because of $f$ being integrable I can write that $\sum \limits_{i=0}^{n-1}(\sup f-\inf f) \Lambda x_{i} < \varepsilon$ for a division of $[a,b]$.
I tried to define $g=|f(x+h)-f(x)|$ and work with that, but it didn't lead me to any smart conclusion. Also I tried to put the limit inside of the integral and to use the derivative, but I'm not sure when I can do that.
Thank you for the help!