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.

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April 14, 2010 - Posted by | Analysis, Measure Theory

7 Comments »

  1. [...] Example of Monotonicity We continue with our example and show that the set function which assigns any semiclosed interval its length has various [...]

    Pingback by An Example of Monotonicity « The Unapologetic Mathematician | April 15, 2010 | Reply

  2. [...] Example of a Measure At last we can show that the set function we defined on semiclosed intervals is a measure. It’s clearly real-valued and non-negative. We already showed that it’s [...]

    Pingback by An Example of a Measure « The Unapologetic Mathematician | April 16, 2010 | Reply

  3. [...] So we’ve identified a measure on the ring of finite disjoint unions of semiclosed intervals. Now we want to apply our extension and completion [...]

    Pingback by Lebesgue Measure « The Unapologetic Mathematician | April 19, 2010 | Reply

  4. [...] as I said when I introduced semiclosed intervals, we could have started with open intervals, but the details would have been messier. Now we can see [...]

    Pingback by Borel Sets and Lebesgue Measure « The Unapologetic Mathematician | April 20, 2010 | Reply

  5. [...] as I said when I introduced semiclosed intervals, we could have started with open intervals, but the details would have been messier. Now we can see [...]

    Pingback by Borel Sets and Lebesgue Measure « The Unapologetic Mathematician | April 20, 2010 | Reply

  6. This item on semiclosed intervals is the first blather I have read. It is so good that I went looking on Amazon for a book by J.A. I was disappointed there, but I plan to read more here.

    Comment by Gary | May 22, 2010 | Reply

  7. I should add that this site crashed my Google Chrome tab twice, the second time on submitting my first comment. The rescue provided by resending the URL worked in both cases.

    Comment by Gary | May 22, 2010 | Reply


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