The Unapologetic Mathematician

Mathematics for the interested outsider

Semiclosed Intervals

Before we go any further, let’s work towards an actual example of a measure. This one, in the long run, will be useful to us.

The underlying space X we’re interested in is the real line. We need to start with a class \mathcal{P} of sets we’re interested in measuring. Specifically, we’re going to take \mathcal{P} to be the class of finite intervals, open on the right and closed on the left. That is, given finite real numbers a<b we consider the interval

\displaystyle\left[a,b\right)=\{x\vert a\leq x<b\}

Such a bounded interval we’ll call “semiclosed”. We’ll also throw \emptyset into \mathcal{P} and let this count as a degenerate sort of semiclosed interval.

Now, given two semiclosed intervals, their intersection is again a semiclosed interval. One possibility is that one interval contains the other, in which case the intersection is the smaller interval. Another possibility is that the intervals are disjoint, in which case their intersection is empty. The last possibility is that they overlap: we consider \left[a,b\right) and \left[c,d\right) with a<c<b<d. Then their intersection is \left[c,b\right), which is a semiclosed interval.

The difference of two semiclosed intervals may or may not be a semiclosed interval. If intervals overlap, as above, then \left[a,b\right)\setminus\left[c,d\right)=\left[a,c\right), and \left[c,d\right)\setminus\left[a,b\right)=\left[b,d\right). If the intervals are disjoint, then the difference is just the original interval. But if \left[a,b\right) contains \left[c,d\right), then the difference is \left[a,c\right)\cup\left[d,b\right). This isn’t a semiclosed interval, but it’s a finite disjoint union of semiclosed intervals.

But we know that these properties are exactly what we need to show that the collection \mathcal{R} of finite disjoint unions of intervals in \mathcal{P} is a ring. We could have started with open intervals or closed intervals, but then we wouldn’t have such a nice ring pop out.

We will define a finite set function \mu:\mathcal{P}\rightarrow\mathcal{R}. For an interval \left[a,b\right), we define \mu(\left[a,b\right))=b-a. For the empty set, we define \mu(\emptyset)=0. This is the function that will be developed into our measure.

April 14, 2010 Posted by | Analysis, Measure Theory | 8 Comments