Transfinite exhaustion

Hi all!
One exercise from Halmos's Measure theory follows. My friend succeeded
to solve it by other means - so I'm not asking for the solution - but I
wonder what he means under transfinite exhaustion, so I wanted put here
the whole context of this expression.
Any reference to a book, site or some proof, which uses this method
would help.
Thanks in advance!
Martin

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.]

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