Why are complex measures not allowed to attain $\infty$ while signed measures are?

Question

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.

Answer 1

As suggested, I’m promoting my first two comments to an answer:

I don’t think there are any immediate problems with what you wrote down. You can certainly allow a complex measure to take value $\infty$ if you define it carefully. I assume it’s mostly because the reason complex measures are interesting is precisely because they form a vector space over $\mathbb{C}$, and most of the time we only need them because the vector space they form correspond to some other vector space. (For example, the space of complex Radon measures on a compact Hausdorff space is the dual space of the space of continuous functions.)

Though it does feel a bit weird to say $i\infty = \infty$ whereas the $-i\infty$ is disallowed. Generally speaking, complex infinity does not distinguish between different directions in the complex plane. This might be another reason why complex measures don’t usually allow infinity. For signed measures, my impression is that they have other uses beyond just considering the space they form, so allowing infinity is more sensible. (And distinguishing between $\infty$ and $-\infty$ is also more sensible to do, given the natural ordering on $\mathbb{R}$.)

Answer 2

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.