How Does Length Extend to General Sets of Real Numbers?

[Artusartos]

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

 

[Whovian]

Yes, but no matter which x and y we choose (and if we add in more variables, such as w and z,) the sum of the lengths is going to be the same. If we take a line segment and chop it up into bits, the sum of the lengths of those bits is always going to be the length of the original segment.

Them using the word "countable" and saying that an open interval is composed of open intervals, however, makes me hesitate on this. I'm not too far into set theory, but countable makes it sound like they're talking about an infinite number of intervals.

 

[mathman]

The point of countable is that any collection of non-overlapping open intervals is finite or countably infinite, so you can add up the lengths.

 

[HallsofIvy]

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".

Your quote above does not say that- it does not say a "countable number of open intervals" in an open interval, it says a "countable number of open intervals" in an open set. So it is referring to finding the length of things like (0, 1)\cup (3, 10). That set would have length (1- 0)+ (10- 3). That is, it is building up general sets in terms of unions of intervals. (Not all sets can be written as countable unions of intervals so there will still be some sets for which we cannot define "length".)

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.

By using the definition given for the length of an open interval: the length of (a, b)= b- a. If you are asking how we can know those intervals, well that depends on exactly how the set itself is given. The point here was to generalize length from intervals to more general sets. For example, the "set of all rational numbers between 0 and 1" is itself countable and so can be written as a countable union of singleton sets- sets containing a single point. Such a set has length 0, of course, so the "set of all rational numbers between 0 and 1" has length 0. And, from that we see that, since the length of the interval (0, 1) is 1, the "set of all irrational numbers between 0 and 1" has length 1 as well.

But, as I said before there will always be sets that cannot be written that way and so have no "length".

I was wondering if anybody could give me an example in order to clarify what this means...

Thanks in advance

 

[Artusartos]

Your quote above does not say that- it does not say a "countable number of open intervals" in an open interval, it says a "countable number of open intervals" in an open set. So it is referring to finding the length of things like (0, 1)\cup (3, 10). That set would have length (1- 0)+ (10- 3). That is, it is building up general sets in terms of unions of intervals. (Not all sets can be written as countable unions of intervals so there will still be some sets for which we cannot define "length".)


By using the definition given for the length of an open interval: the length of (a, b)= b- a. If you are asking how we can know those intervals, well that depends on exactly how the set itself is given. The point here was to generalize length from intervals to more general sets. For example, the "set of all rational numbers between 0 and 1" is itself countable and so can be written as a countable union of singleton sets- sets containing a single point. Such a set has length 0, of course, so the "set of all rational numbers between 0 and 1" has length 0. And, from that we see that, since the length of the interval (0, 1) is 1, the "set of all irrational numbers between 0 and 1" has length 1 as well.

But, as I said before there will always be sets that cannot be written that way and so have no "length".

Thanks a lot