I happened to get a glimpse of an exercise and a claim in it soon beset me:
I have no problem about the exercise itself. What puzzled me is that I cannot find any justification for the claim that $(+\infty)(-\infty)$ is undefined. What's wrong with just defining the product to be $-\infty$? Is it a typo of the text, or did I miss anything in understanding the undefinedness of this product? I hope someone can please help me clarify it. Thank you. PS: this exercise follows a section discussing measure theory. PS2: by "$\mathcal{E}$-measurable", it means the co-domain is $\bar R$.
Here is the principle behind the notion of "(un)determined" operations involving $0$ and $\pm \infty $.
Let $$ \overline{\mathbb R} = \mathbb R ∪ \{+\infty , -\infty \}, $$ and let $\odot$ be any mathematical operation such as sum, difference, product, division or exponentiation. Given $\alpha $ and $\beta $ in $\overline{\mathbb R}$, the question I want to address here is what does it mean to say that $\alpha \odot\beta $ is (un)determined.
For the sake of argument suppose that $\{x_n\}_n$ and $\{y_n\}_n$ are sequences of real numbers such that $$ \lim_{n\to \infty } x_n=\alpha , \quad \text{and} \quad \lim_{n\to \infty } y_n=\beta . $$
One may then ask what is the value of $ \displaystyle\lim_{n\to \infty } x_n\odot y_n $?
It is often the case that the answer to this question cannot be found based only on the information given! For example, if $x_n=1/n$, $x'_n=2/n$, and $y_n=n$, then $$ \lim_{n\to \infty } x_n=0=\lim_{n\to \infty } x'_n, \quad \text{while} \quad \lim_{n\to \infty } y_n=\infty . $$ However $$ \lim_{n\to \infty } x_ny_n = 1 \neq 2 = \lim_{n\to \infty } x_n'y_n. $$
Does this say that $0\cdot \infty =1$, or would it be $2$? Well, I guess this says that $0\cdot \infty $ is undetermined!
Based on this, one may give the following formal definition:
Definition. If $\alpha,\beta \in \overline{\mathbb R}$, and if $\odot$ is any mathematical operation, we say that $\alpha \odot\beta $ is determined, and that its value is $\gamma $, if for every sequences $\{x_n\}_n$ and $\{y_n\}_n$ (of standard real numbers) such that $$ \lim_{n\to \infty } x_n=\alpha , \quad \text{and} \quad \lim_{n\to \infty } y_n=\beta , $$ one has that $\displaystyle\lim_{n\to \infty } x_n\odot y_n=\gamma $. Otherwise we say that $\alpha \odot\beta $ is undetermined.
According to this, it is clear that $(+\infty )\cdot (-\infty )=-\infty $, and that $0\cdot \infty $ is undetermined.
Nevertheless, there are certain areas of Math in which it might be sensible to adopt special conventions.
In Integration Theory, as mentioned by @DanielFischer, it is sensible to define $0\cdot \infty =0$, because the integral of the zero function over an infinite measure space is equal to zero. Another example is Algebra, where the definition of a polynomial as $$ p(x) = \sum_{n=0}^d a_nx^n, $$ only makes sense if one adopts the convention that $x^0=1$, the case $x=0$ included!
PS: I am well aware that the level of my answer is not quite on par with the question. But it turns out that the present answer, which is certainly what the 'experts' have in mind, is not too often spelled out and perhaps Math begginers might benefit from it!