What exactly happens when we assume the convention $0 \times \infty := 0$?

Question

So I know that in measure theory we consider functions with values $+/- \infty$ because we want to consider limits of functions, since Lebesgue integral is made to have some nice limit-preserving properties.

1. Is there some other reason for considering infinite-valued functions?

Also, what exactly happens when we assume that $0 \times \infty := 0$ and $\infty \times 0 := 0$ in the definition of integrals of simple functions? What we get is that $[-\infty, \infty]$ is not even a group with respect to multiplication or addition, since $\infty \times -\infty$ and $\infty -\infty$ are not well defined. I understand that it's a very useful convention, but it seems potentialy risky.

My thoughts: what breaks down is "continuity" of multiplication and addition when considering $[-\infty, \infty]$ with it's usual topology. That is, some important (but maybe not really?) properties of continuity fail:
For multiplication: we would like to say that multiplication is "continuous" only if it implies that for every pair of convergent to some (possibly infnite) values sequences $\{x_n\}$, $\{y_n\}$ the $\lim_{n \rightarrow \infty} (x_n \times y_n)$ is well definied and $\lim_{n \rightarrow \infty} (x_n \times y_n)= \lim_{n \rightarrow \infty} x_n \times \lim_{n \rightarrow \infty} y_n$. This fails.
For addition: we would like to say that addition is "continuous" only if it implies that for every pair of convergent to some (possibly infnite) values sequences $\{x_n\}$, $\{y_n\}$ the $\lim_{n \rightarrow \infty} (x_n + y_n)$ is well definied and $\lim_{n \rightarrow \infty} (x_n + y_n)= \lim_{n \rightarrow \infty} x_n + \lim_{n \rightarrow \infty} y_n$. This fails as well.
So we have to "watch out". But from my experience we don't seem to.

2. So why isn't it the case that we have to be very carefull with this convention? From these bad properties of multiplication and addition I would suspect that most proofs in measure theory dealing with Lebesgue integral (and the limits theorems of integrals in particular) would have to go like (roughly speaking, since I want to be as general as possible) "Let's prove the theorem for $\mathbb{R}$ first, and then separately for the pathological case of $+/-\infty$". But it's not at all a common theme in measure-theoretic proofs in my experience. Generaly, why can we dispense with such a carefull attitude?

3. What are some other important properties that break down when assuming this convention? Can we even classify "exactly what breaks down comapred to $\mathbb{R}$"?

I couldn't find any satisfying answers for these questions on the forum.

Answer 1

Set theory offers a scenario where zero times an infinite cardinality is zero. To wit, take the cross-product of any infinite set with the empty set, and the result is the empty set; then the multiplication rule for set cardinalities forces the claimed product.

However, infinite set cardinalities are distinct entities from what is called infinity in other areas of analysis, so this definition cannot be carried over.

Answer 2

I remember reading in Walter Rudin's RCA that with this convention, the commutative, associative, and distributive laws are shown to hold in the range $[0,\infty]$ "without any restriction."