In fact the total variation of a Banach valued vector measure is always a measure, as you are saying. What can happen is that $\|\mu\|$ has "heavy atoms", that is, measurable, non null sets $A\in\mathcal{A}$ such that any measurable $B\subset A$ is either $\|\mu\|$-null or has infinite $\|\mu\|$ measure. The space $\Omega$ itself may result a "heavy atom", that is the image of $\|\mu\|$ as a function is reduced to $\{0,+\infty\}$. Maybe the lecture notes wanted to rule out these somehow pathological situations.

In fact the total variation of a Banach valued vector measure is always a measure, as you are saying. What can happen is that $\|\mu\|$ has "heavy atoms", that is, measurable, non null sets $A\in\mathcal{A}$ such that any measurable $B\subset A$ is either $\|\mu\|$-null or   $\|\mu\|$-infinite. The space $\Omega$ itself may result a "heavy atom", so that the image of $\|\mu\|$ as a function is reduced to $\{0,+\infty\}$. Maybe the lecture notes wanted to rule out these somehow pathological situations.