On a measure space $(\Omega, \mathcal{F}, \mu)$, in Rudin's Real and Complex Analysis, with some modification of notations:
If there is a set $A \in \mathcal{F}$ such that $\mu(E) = \mu(A \cap E) $ for every $E \in \mathcal{F}$, we say that $\mu$ is concentrated on $A$.
I was wondering if according to the definition, it is true that $\mu$ is concentrated on $A \in \mathcal{F}$ if and only if $\mu(A) = \mu(\Omega) $? If yes, why was the definition made more complicated than what it is?
Thanks and regards!