Our textbook states:
The length l(I) of an interval I is defined to be the difference of the endpoints of I if I is bounded, and infinity if I is unbounded. Lenght is an example of a set function, that is, a function that assosiates an extended real number to each set in a collection of sets. In the case of length, the domain is the collection of all intervals. In this chapter we extend the set function length to a large collection of sets fo real numbers. For instance, the "length" of an open set will be the sum of the lengths of the countable number of open intervals of which it is composed."
I don't understand this last sentence.
1) How can there be a "countable number of open intervals" in an open interval? For example, if we have $(0,2)$, I can choose any two numbers x,y such that $0 < x,y <2$ and create an open interval from them, right? So I don't understand what they mean by a "countable number of open intervals".
2) The text is telling us the the "'length' of an open set will be the sum of the lengths of the countable number of open intervals of which it is composed". But how can we know the length of those "countable number of open intervals of which it is composed.
I was wondering if anybody could give me an example in order to clarify what this means…
Thanks in advance
Best Answer
1) The phrase "of which it is composed" seems to be intended to mean that you write the open set $U$ as $$\tag1 U=\bigcup_{i\in S}(a_i,b_i)$$ Where $(a_i,b_i)\cap (a_j,b_j)=\emptyset$ for $i\ne j$.
2) In (1) above, the length of $(a_i,b_i)$ is $b_i-a_i$. Then the author wants to define $$l(U)=\sum_{i\in S} (b_i-a_i).$$ Of course, one needs to show that this s well defined