I am trying to understand how continuity of measures relates to the definition of continuity in topological sets : Every open set in range corresponds to an open set in domain.
A real valued set function $\mu$ defined on a $\sigma-field$ $\mathcal{F}$ is continuous from below: $A_1, A_2, \cdots A_n\cdots \in \mathcal{F}$ and $A_n \uparrow A \implies \mu(A_n) \to \mu(A)$
Can this also be stated as : For some value $\delta< \mu(A)$, every interval $(a,b) \subset (\delta,\mu(A)]$ has a corresponding collection of sets $A_m, A_{m+1} \cdots A_k \in \mathcal{F}$, i.e. $A_m,A_{m+1}\cdots A_k$ are consecutive sets.
Does this also imply that for every unique value $x \in (\delta,\mu(A)]$ there exists an $A_k \in \mathcal{F}$ such that $\mu(A_k)=x$