But isn’t the issue that coastlines have a fractal nature? That depending on your resolution, you could have a finite or infinite length of a coastline? In which case measurement is hard to define.

Talking about integrals, the fun part is that even with a coastline of indeterminate length, the area of a continent is easy to define to arbitrary precision - you can just define an integral that’s definitely inside the area and one that’s definitely outside the area, and the answer is between those two.