I'm reading A First Look at Rigorous Probability Theory and I'm struggling to understand what the outer measure defined below would give as an "output." As seen below, the outer measure $\mathbf{P}^*(A)$, for any subset $A \subseteq \Omega$, is defined to be the infimum of sums of $\mathbf{P}(A_i)$, where $\{A_i\}$ is any countable collection of elements of the original semialgebra $\mathcal J$ whose union contains $A$. What does the infimum of sums here mean? Is it be the smallest probability amongst all probabilities $\mathbf{P}(A_i)$ or what specifically? We know that for $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$, $\inf (A+B) = 5$. What would now be $\inf (\mathbf{P}(A)+\mathbf{P}(B)) = \cdots$ be? Sorry for the naive question and thanks in advance.
The notation of the book is not so good. A cleaner notation is
$$ P^*(A):=\inf \left\{\sum_{k=1}^{\infty }P(A_k):A\subset \bigcup_{k=1}^{\infty }A_k,\, \{A_k\}_{k\in\mathbb N}\subset\mathcal{J}\right\} $$