Let $L_n : \mathbb{R}^m \to \mathbb{R}^m $ be a sequence of lipschitz function with lipschitz constants $K_n \le \frac{1}{2}$ and $\lim_{n \to \infty}L_n = 0$. Let $f: \mathbb{R}^m \to [0, \infty)$ be a measurable function. How can one show that:
$$\lim_{n \to \infty}\int_{\mathbb{R}^m}|f(x)-f(x+L_n(x))|dx = 0$$ holds true? Does anyone has a hint?