Title: Certified Adversarial Robustness Under the Bounded Support Set. Abstract: Deep neural networks (DNNs) have revealed severe vulnerability to adversarial perturbations, beside empirical adversarial training for robustness, the design of provably robust classifiers attracts more and more attention. Randomized smoothing method provides the certified robustness with agnostic architecture, which is further extended to a provable robustness framework using $f$-divergence. While these methods cannot be applied to smoothing measures with bounded support set such as uniform probability measure due to the use of likelihood ratio in their certification methods. In this paper, we introduce a framework that is able to deal with robustness properties of arbitrary smoothing measures including those with bounded support set by using Wasserstein distance as well as total variation distance. By applying our methodology to uniform probability measures with support set $B_{2}(O,r)$, we obtain certified robustness properties with respect to $l_{p}$-perturbations. And by applying to uniform probability measures with support set $B_{\infty}(O,r)$, we obtain certified robustness properties with respect to $l_{1},l_{2},l_{\infty}$-perturbations. We present experimental results on CIFAR-10 dataset with ResNet to validate our theory. It is worth mentioning that our certification procedure only costs constant computation time which is an improvement upon the state-of-the-art methods in terms of the computation time.