@NateEldredge. $E$ is usual Lebesgue integral. Yes you are right, if the lengths of atoms of partition are same then it can be shown $$\underset{t\in J}{\sup}E(|G_{n}(\cdot, t)-f|^{p})\leq 2\underset{|h|\leq \lambda^{n}}{\sup}E(|f(x+ h)-f(x)|^{p})$$. But in the case of different lengths I have got some problem

Yes, you are right. I agree with your comment. During my study I faced such problem and I have put this problem without any changes. Thank you