Let $A=\bigcup_{i=1}^n (a_i,b_i]\subset(0,1]$ where $(a_i,b_i]$ are disjoint. I define a function $P (A)=\sum_{i=1}^n (b_i-a_i) $. I wish to prove that $P (A )$ is well defined in the sense that if there is some other way to represent $A $ as a finite union of disjoint intervals the same value of $P (A) $ results.
My book claims that this follows from the fact that if $A,B $ are disjoint then $P (A\cup B) =P (A)+P (B) $. I don't quite see how though.
I think I have discovered the answer: Given any representation of $A$ as a finite disjoint union of intervals (all of the type (a,b]) the Riemann integration of the indicator function will be the sum of the interval lengths and hence $P(A)$. For any other such representation since the Riemann integration of the indicator function would not change so $P(A)$ would not either.