While studying the principles of mathematical analysis (Rudin) alone, I faced one sentence I couldn't understand.
In 11.2 Definition (page301), it explains the "additive" and "countably additive" set function.
Below the definition, it says "we shall always assume that the range of $\phi$ does not contain both $+\infty$ and $-\infty$; for if it did, the right side of (3) could become meaningless. Also, we exclude set functions whose only value is $+\infty$ or $-\infty$".
I'm curious about the reason why the right side of (3) could become meaningless if the range of $\phi$ contains both $+\infty$ and $-\infty$.
In addition, I'm curious about why we need additional condition of $\vert(\phi B)\vert < +\infty$ in (9) compared to (3).
For $(3)$ we need the range to be shy of at least one of $\pm\infty$, as one can stumble onto the case when $\phi(A)=\infty$ and $\phi(B)=-\infty$, in which case $\phi(A)+\phi(B)$ is not defined.
For $(9)$, note that $A\cup B$ can be written as the union of the disjoint sets $A\setminus B$ and $B$, so that by $(4)$ one has $\phi(A\cup B)=\phi(A\setminus B)+\phi(B)$. So, if $A\supset B$, $(9)$ readily follows. In addition, one requires $\phi(B)<\infty$, as otherwise, by $(8)$ it would mean $\phi(A)=\infty$, and we have $\infty-\infty$ scenario again.