2
$\begingroup$

My interest is more about getting back to Kansas than it is about following the Yellow Brick Road; I am trying to understand the ultimate logic, not the abstractions along the way.

I have spent a great deal of my life dealing with probabilistic methods, though I was trained as an engineer, not a mathematician. When it comes to measurable functions and sigma algebras, I don't quite see the magic.

One book gave an abstract example of the set of sets $\{\emptyset, \{a,b\},\{c,d\},\{a,b,c,d\}\}$meets the tests since the unions of any elements leads to another element, as does the intersections. But, I suppose, $\{\emptyset, \{a\}, \{a,b\},\{c,d\},\{a,b,c,d\}\}$does not, since $\{a\}\cup\{c,d\}$ is not a member, correct?

Now, let's get away from finite sets and abstractions like this, and looking at the real number line in one dimension, what is accomplished by the fact that a countably infinite set of unions or intersections of open sets is also a member of the set of open sets on the real line?

What would go wrong if they were closed sets? It seems intuitively that, on the real line, unions and intersections of closed sets are also closed sets. One could, for example, take the intersection of all closed sets of the form $[q,q+1]$ (q an integer) which would deliver up the set of integers, as all the intervals would have exactly one point in common with one other interval. Perhaps this is abusive notation, but all of these are closed sets if one can write the interval $[1,1]$ meaning the point at 1. Or is that exactly what is trying to be avoided?

Is it simply that open sets that are non-empty have measure, and by narrowing ourselves to those it just makes measure theory work out, in part because the opposite of what would happened with closed sets occurs - i.e., the union of all open sets of the form $(q,q+1)$is the complement of the intersection of the closed sets: it is the real number line missing countably infinite entries (the integers) - but has well defined measure for a finite collection of such abutting sets?

Improve this question
$\endgroup$
  • 1
    $\begingroup$ Both the measure on $\Bbb R$ and its topology are defined by intervals. Actually, a closed interval can also be written as a countable intersection of open intervals and vica versa, so it doesn't matter which we start off. In measure theory we want to perform sum of countably infinitely many measurable sets. The need for introducing $\sigma$-algebras as a 'support' for measures is because it turns out (following e.g. by Banach-Tarski's sphere duplication theorem) it's not possible to measure all subsets of the Euclidean spaces. $\endgroup$ – Berci Sep 15 '18 at 23:36
  • $\begingroup$ Some passages in your question, like "the intersection of all closed sets of the form $[q,q+1]$ ($q$ an integer) which would deliver up the set of integers," suggest that you have not correctly understood what is meant by the intersection of a family of sets. $\endgroup$ – Andreas Blass Sep 15 '18 at 23:46
  • 1
    $\begingroup$ Interesting question, similar to this one in CrossValidate.SE. $\endgroup$ – Antoni Parellada Sep 16 '18 at 0:33
  • $\begingroup$ The reference to Cross-Validated is quite useful, but as noted in the comments it still does not explain quite what would "break" if countably infinite intersections and unions weren't required as part of the sigma algebra. $\endgroup$ – eSurfsnake Sep 16 '18 at 4:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.