## An Example of Monotonicity

We continue with our example and show that the set function which assigns any semiclosed interval its length has various monotonicity properties.

First off, let be a finite, disjoint collection of semiclosed intervals, all of which are contained in another semiclosed interval . Then we have the inequality

Indeed, we can write , , and without loss of generality assume that . Then our hypotheses tell us that

and thus

On the other hand, if is a closed interval contained in the union of a finite number of bounded open intervals , then we have the *strict* inequality

We can rearrange the open intervals by picking to contain . Then if we have and we can discard all the other sets since they only increase the right hand side of the inequality. But if , we can pick some containing . Now we repeat, asking whether is greater or less than . Eventually we’ll have a finite collection of satisfying , , and . It follows that

What does this have to do with semiclosed intervals? Well, if is a countable sequence of semiclosed intervals that cover another semiclosed interval , then we have the inequality

If , then this is trivially true, so we’ll assume it isn’t, and let be a positive number with . Then we have the closed set . We can also pick any positive number and define .

Now is smaller than , and each is larger than the corresponding , and so we find that is a closed interval covered by the open intervals . But the Heine-Borel theorem says that is compact, and so we can find a finite collection of the which cover . Renumbering the open intervals, we have

and our above result tells us that

Since we can pick and to be arbitrarily small, the desired inequality follows.

[...] intervals is a measure. It’s clearly real-valued and non-negative. We already showed that it’s monotonic, and this will come in handy as we show that it’s countably [...]

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