I was reading Evan Chen's Napkin, and ran across this example in the section focusing on pre-measures:
Based on my understanding, the fact that we're working with finitely many rectangles and their complements and unions ensures that we can unambiguously define area (and we do so later with the pre-measure of rectangle $\mu_0([a_1,a_2] \times [b_1,b_2] = (a_2-a_1)(b_2-b_1)$).
My question: does this break down for countably many rectangles? And if so why? Essentially I want to know if we can define a rectangle in two different ways, one of which may be using countably many rectangles, and have the two areas be different.
Some ideas I had were with taking a countably infinite number of rectangles with area 0 and 'stitching' them together to get a rectangle with a positive area, but I suspect that this wouldn't work without using uncountably many rectangles. Another idea was to do something similar to Zeno's paradox where to create a rectangle, I would create a rectangle with half its width, and then with half of that width, so on and so forth. But I see no reason why this should cause any
You may get an ambiguity if you accept either rectangles with negative area, or subtraction of areas. The sum of a series depends upon the ordering of terms iff the series is not absolutely convergent. But a series of positive or null terms, if convergent, is absolutely convergent, so you cannot create any "paradox" by summing only positive or null values.
Accepting negative values is probably the case in the book you refer, because it talks about an algebra.
There can be an ambiguity such as this one: take rectangles with length $\frac 1 n$ for $n \in \mathbb{N}_{>0}$, and heigth $1$. Consider adding rectangles for $n$ even and subtracting rectangles for $n$ odd. Then, depending upon the order in which you add and subtract those rectangles, you may get any area in $[-\infty, \infty]$ - where a negative area is when the sum, beginning at abscissa $0$, gets to a negative abscissa.