How can I show that if $f\in L^p(a, b)$ then $\lim_{t\to 0^{+}}\int_{a}^b |f(x+t)-f(x)|^p\ dx=0$..

Question

can anyone help me show that if $f\in L^p(a, b)$ then $$ \lim_{t\to 0^{+}}\int_{a}^b|f(x+t)-f(x)|^p\ dx=0.$$ Thanks, any help will be useful..

Answer 1

Let $\tau_h f(x) = f(x+h)$ denote the translation operator. If $f$ is a continuous compactly supported function, the desired assertion follows from the dominated convergence theorem. Otherwise, there exists a continuous compactly supported function $g$ with $\|f - g \|_p < \epsilon$. The translation invariance of Lebesgue measure implies $$\|f - \tau_h f\|_p \le \|f - g \|_p + \|g - \tau_h g \|_p + \|\tau_h g - \tau_h f \|_p$$ $$\le 2\|f - g \|_p + \|g - \tau_h g \|_p \le 3\epsilon$$ for small enough $h$.