Let $U$ be a $[0,1)$-valued random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$, $k\in\mathbb N$, $I:=\{0,\ldots,k-1\}$ and $\zeta$ denote the counting measure on $(I,2^I)$. Does $S:=\lfloor kU\rfloor$ admit a density wit respect to $\zeta$? Clearly, by definition, $$\operatorname P\left[S=i\right]=\operatorname P\left[U\in\left[\frac ik,\frac{i+1}k\right)\right]\tag1$$ for all $i\in I$. If the desired result is wrong in general, are we able to impose assumptions on the distribution of $U$ such that it holds true?
Remark: In my application, $U$ is essentially distributed to a mixture distribution $p\mathcal U_{[0,\:1)}+(1-p)\mathcal N_{\mu,\:\sigma^2}$ for some $p\in(0,1)$, $\mu\in[0,1)$ and $\sigma>0$. Please take note of my related question for some context: Does the transition kernel of my application fit into this theoretical setting?.