I saw another question similar to this one but I'm not satisfied by answers. Here I changed the question to clearify the point I am interested in. I study Measure Theory for Real & Complex Analysis and I wonder why in the following setting
\begin{equation} \mu: \mathcal{M} \rightarrow \mathbb{C} \end{equation}
\begin{equation} \mu(E) = \mu_r(E) + i\mu_i(E) \end{equation}
where $\mu$ is a complex measure and $\mu_r$ & $\mu_i$ are signed measure some writers (like Folland if I remember correctly) does not let $\mu$ to attain at most one of $\infty$ or $-\infty$. In the general definition for signed measures we can let signed measures to attain at most one of the plus or negative $\infty$.
For example, is the following setting not meaningful or not sensible?
\begin{equation} \mu: \mathcal{M} \rightarrow \mathbb{C}\cup \{\infty\} \end{equation}
\begin{equation} \mu(E) = \mu_r(E) + i\mu_i(E) \end{equation}
where $\mu_r$ & $\mu_i$ are signed measures that does not attain $-\infty$, i.e. they are signed measures which attain values in $\mathbb{R} \cup \infty$, and we defined $\mu(E) = \infty$ whenever $\mu_r = \infty$ or $\mu_i = \infty$. Here I rely on following artihmetical definitions in a formal way:
\begin{equation} a + i\infty = \infty = \infty + ai \end{equation}
where $-\infty$ is not considered.
EDIT: Typo in the last equation.
Best Answer
Complex measures are a linear space over complex numbers. In order to maintain that with your extension, you would also need to define $a\cdot \infty$ and $a + \infty$, $\infty + \infty$, etc. But there is no good definition of those that maintains $\mathbb{C} + \{\infty\}$ as a linear space. For instance, what would $\infty - \infty$ be?
Edit:
With your definition, you might have a valid extended complex measure, but such that multiplying that measure by a complex constant does not yield a valid extended complex measure. Valid measures aren't closed under multiplications by a constant.
For instance: take the space of natural numbers. Even numbers have measure 1, odd numbers have measure $i$. It's a valid measure by your definition.
But now if you multiply this measure by $1 + i$, it's no longer a valid measure. Now even numbers have measure $1 + i$, odd numbers have measure $-1 + i$, and the sum of real parts for the whole space isn't absolutely convergent.