Consider two measurable spaces $X_n = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu_n)$, $n\in\mathbb{N}$, and $X = (\mathbb{R}^m,\mathcal{B}(\mathbb{R}^m),\mu)$. Morever, consider the product spaces $X_n^{q} = (\times_{i=1}^q\mathbb{R}^m,\otimes_{i=1}^q\mathcal{B}(\mathbb{R}^{m}),\mu_n^{\otimes q})$ and $X^{q} = (\times_{i=1}^q\mathbb{R}^m,\otimes_{i=1}^q\mathcal{B}(\mathbb{R}^{m}),\mu^{\otimes q})$ where $\mu_n^{\otimes q} = \underbrace{\mu_n\otimes \cdots \otimes\mu_n}_{q}$ and $\mu^{\otimes q} = \underbrace{\mu\otimes \cdots \otimes\mu}_{q}$, respectively.
Let $$ D(\mu,\nu) = \sup_{\text{all closed balls } B \text{ of }\mathbb{R}^m} \left|\mu(B)-\nu(B)\right| $$ be the discrepancy metric between measures $\mu$ and $\nu$.
My question. If $D(\mu,\mu_n)\le c_n$, then there exists $K>0$ independent of $n$ such that $$ \left|\mu^{\otimes q}(S)-\mu_n^{\otimes q}(S)\right|\le K c_n $$ for all $S\in\otimes_{i=1}^q\mathcal{B}(\mathbb{R}^{m})$?