Re: Transfinite exhaustion

Martin wrote:
> One exercise from Halmos's Measure theory follows. [...] I
> wonder what he means under transfinite exhaustion, so I wanted put here
> the whole context of this expression.
>
> Halmos: Measure theory, p.174, exercise 2
>
> If (S,mu) is a sigma-finite, non atomic measure ring, and if E0 in S,
> then, for every extended real number a with 0 <= a <= mu(E0) there
> exists an element E is S such that E subset E0 and mu(E)=a.
> (Hint: Since the case a=infty is trivial, there is no loss of
> generality in assuming that mu(E0)<infty. The desired result follows by
> a transfinite exhaustion process. The method is similar to the one used
> in proving that any two points in a complete, convex metric space may
> be joined by a segment, and in fact the present assertion is a special
> case of this general theorem in metric geometry.)
>
> [If I'm not mistaken, the theorem Halmos mention's is known as Menger's
> theorem.]


My guess is that transfinite exhaustion refers to Zorn's lemma. Halmos
often uses such phraseology in reference to induction through ordinals
or its equivalent (e.g., ZL). Does the proof you know use ZL?

-S.J. Herschkorn
Tutor on the Internet and in Central New Jersey and Manhattan

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