$\infty + \infty = \infty$?

Question

(The context is a measure-theoretic one.) I know that $\infty - \infty$ is indeterminate, but what about $\infty + \infty = \infty$?

It seems this statement is true and if I input it into Wolfram Alpha it says it's true. But here it says it is not true and that this equation is also indeterminate. Is there some extra condition in the linked post that I am overlooking...surely $\infty + \infty = \infty$ always?

Answer 1

The linked post is about $\infty$ the point at infinity of the Riemann sphere/the complex projective line.

In that situation (like if $\infty$ denotes the point at infinity of the real projective line), $\infty + \infty$ is indeterminate, since $z_n$ and $w_n$ can approach $\infty$ in such a way that $z_n + w_n$ converges to any point of the projective line, or not at all. The addition cannot be continuously extended to $\widehat{\mathbb{C}}\times \widehat{\mathbb{C}}$ (resp. $\widehat{\mathbb{R}}\times \widehat{\mathbb{R}}$).

In the context of measure theory, $\infty$ is "positive infinity", which completes the non-negative half-line to $[0,\infty]$, and in that setting, the addition can be continuously extended to $[0,\infty]\times[0,\infty]$ by setting $x+\infty = \infty = \infty + x$ for all $x$, so there $\infty + \infty$ is not indeterminate.

It would still be better to use the symbol $+\infty$ to avoid ambiguity.

Answer 2

A good question might be what does $+$ mean here?

At a basic level, $\infty$ is not a number and can't be added like numbers. Without further clarification, this is how I treat infinities that I see for the first time.

In the real line, we might very well mean "arbitrarily large and positive" by $\infty$ (and "arbitrarily large [in absolute value] and negative" by $-\infty$). This is sort of how infinity gets played with in standard calculus courses. So here, $\infty + \infty$ is just another arbitrarily large and positive number, so we might say that $\infty + \infty = \infty$. On the other hand, $\infty + -\infty$ is indeterminate, because arbitrarily positive plus arbitrarily negative doesn't really mean anything.

Measures are typically functions $\mu:X \longrightarrow \mathbb{R}\cup\{-\infty\}\cup\{\infty\}$, where $\infty$ and $-\infty$ might be interpreted as being "bigger than every positive number" and "smaller than every negative number", respectively. [There are more formal definitions of the extended reals, such as a two point compactification with the order topology]. Because of this, the standard operations on the reals can be partially extended to results of measures, and it isn't hard to show that having $\infty + \infty = \infty$ can be made both well-defined and meaningful here.

For contrast, some people might consider a complex measure $\nu: X \longrightarrow \mathbb{C}\cup\{\infty\}$ [aka the Riemann sphere, a 1 point compactification]. The complex numbers aren't ordered, which is why we don't care to add an additional symbol $-\infty$. In this case, we can't say that $\infty + \infty = \infty$ in any meaningful way, because now $\infty$ means colloquially "farther than any number from $0$."

So in short, with a standard measure, you can feel confident saying $\infty + \infty = \infty$ if you really want. But in many cases (and usually by default) people don't treat infinity like a number and thus don't try to add them together.