Why is it possible to write integral if $\int_A f^+ d\mu$ or $\int_A f^- d\mu$ $< \infty$, but for $f$ to be $\mu$-integrable requires both be finite?
That is,
Integral of $f$ over $A$ given either finite is:
$$=\int_A f^+ d\mu-\int_A f^- d\mu$$
But $f$ is $\mu$-integrable, that is, $f \in L^1(A;\mu)$, if both are finite.
I'm not 100% sure what you're asking, but here's an attempt:
It all comes down to $\infty-\infty$ not being well defined.
For a non-negative function $g$, $\int gd\mu=\infty$ makes sense.
However, if you have both $\int f^+d\mu=\infty=\int f^-d\mu$, then $$ \int fd\mu =\int f^+d\mu-\int f^- d\mu=\infty-\infty $$ and there's no meaningful way to assign a value to this.
Moreover, all the integration theorems (Dominated Convergence, etc) only work when the function in question is integrable