Is it consistent (with ZFC) that there is a cardinal $\kappa$ and $m_1$ and $m_2$ two measures on $\kappa$, both $\kappa$-additive, such that
- $m_1$ is atomless, and
- $m_2$ has an atom?
In that case, the existence of $m_1$ implies that the continuum is at least as large as $\kappa$ and the existence of $m_2$ implies that $\kappa$ is inaccessible. That would mean a radical failure of CH. Of course, we know from Easton's Theorem that the size of the continuum can be "almost" anything. Even then, I am curious to know if $m_1$ and $m_2$ can coexist.