Why does infinity have a tradeoff with negative numbers? In Terence Tao's Measure Theory, he writes that

If one wants to keep as many useful laws of algebra as one can, then one can add in [positive] infinity, or have negative numbers, but it is difficult to have both at the same time.

Thus, he says, we develop two overlapping types of measure and integration theory on $[0,+\infty]$ and $(-\infty,+\infty)$. Why does this tradeoff exist? I don't see how negative numbers would mess anything up.
 A: Maybe "negative numbers" doesn't exactly tell you what the issue is. The issue is subtraction. If you don't have negative numbers or subtraction, then there's no issue. The arithmetic behaves predictably and in a well-defined manner:
\begin{align}
a + (b + c) &= (a + b) + c \\
a(b + c) &= ab + bc \\
\infty + a &= \infty \\
\infty \cdot a &= \infty \quad (a \ne 0) \\
\infty \cdot 0 &= 0 \\
\text{etc.}
\end{align}
With subtraction then the naïve system of rules no longer is well-behaved:
$$ \infty = \infty \cdot 1 = \infty \cdot (2 - 1) = \infty - \infty, $$
$$ 0 = \infty \cdot 0 = \infty \cdot (1 - 1) = \infty - \infty. $$
So you get contradictory statements.

I will say that in my area, tropical geometry, it is common to work with the algebraic system (semiring) $(\mathbb{R} \cup \{\infty\}, \oplus, \otimes)$. where
$$ a \oplus b = \min\{a,b\} \text{ and } a \otimes b = a + b. $$
Notice that there's no way to subtract with respect to $\oplus$ so defining $a \oplus \infty = a$ and $a \otimes \infty = \infty$ makes sense and doesn't lead to any contradictory statements. ($\infty$ acts as the "zero of this semiring.)
