Wednesday, January 24, 2007

An easy analysis exercise (+philosophy)

I'm (unfortunately) not teaching a class on analysis, but if I were, I'd assign problems like this (as a warm-up, of course :)

Construct a family of measurable functions whose pointwise supremum is nonmeasurable. [Answers + discussion welcome in comments.]

This very simple exercise illustrates one of the key intuitions about measurability: the only way to obtain non-measurable objects from measurable ones is by some "uncountable process" -- uncountable unions, uncountable suprema, etc. Of course, the very construction of non-measurable sets depends on the axiom of choice, which is equivalent to the well-ordering principle, which manifests itself as Hausdorff's Maximality Principle or Zorn's Lemma; these all enable (uncountably) transfinite induction.

This post was prompted by my reading up on empirical process theory (just when I thought I was starting to get a solid grasp of it, I've discovered whole new oceans) -- mainly stuff by David Pollard and Michel Talagrand. Since uniform convergence involves taking suprema over function families, great care must be taken to ensure measurability (or outer measures must be used, but these create problems of their own). I hope to post on these topics in greater depth when I have time.

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