Assume that $m$ is an atomless probability measure on $\mathbb{R}^{d}$. Let $\left( X_{1},\ldots ,X_{d}\right) $ be a random vector with law $m$. Are the marginal cumulative distribution functions of $\left( X_{1},\ldots ,X_{d}\right)$ continuous? I know how to see this when $d=1$ but I have had a hard time with the general case.
Related with this. Let $F$ be a cumulative distribution function on $\mathbb{R}^{d}$. Can we descompose $F$ as $F=F_{c}+F_{d}$, where $F_{c}$ is a continuous cumulative distribution function and $F_{d}$ is a pure atomic cumulative (i.e., discrete) distribution function? Again, I know that this is true for the case $d=1$.
Thanks for any advice that you can give.
If $X_1=0$ with full probability and the CDF of $X_2$ is continuous then the distribution of $(X_1,X_2)$ is atomless.