Baire Category, Probabilistic Constructions and Convolution Squares

There are several reasons for using the Baire category theorem when we
seek examples of particular types of behaviour.
The first is practical. Although any Baire category argument can obviously
be replaced by a direct argument, if there are several properties
involved, each of which involves countably many conditions the direct argument
may require quite a lot of notation and careful interlocking of several
inductions. Such arguments are not hard to write (and indeed may give the
author some pleasure), but are may be hard to read.
The second argument is that a property which holds quasi-always is, in
some sense, generic. The next exercise shows that we must not press this
argument too far.
Exercise 2.8. The following is a well known procedure for constructing ‘Cantor
sets’. Let E0 = [0, 1] and let ζ1, ζ2, ... be a sequence of real numbers with
0 < ζj < 1. At the nth stage En is the union of 2 n disjoint closed intervals
I(r,n) all of the same length. We define En to be the union of the 2 n+1 disjoint
closed intervals formed by removing an open interval J(r,n) of length
ζn times the length of the initial interval I(r,n) from the centre of I(r,n).
(Thus if I(r,n) = [cr,n −δn,cr,n +δ] we take J(r,n) = (cr,n −ζnδn,cr,n +ζnδn)
and
2
En+1 =
n
(I(r,n) \ J(r,n).
r=1
(i) Explain why ζ = ∞ n=1 ζn is well defined. Show that ζ can take any
value subject only to the condition 1 > ζ ≥ 0.
(ii) Show that E = ∞ n=1 En is a closed nowhere dense set without isolated
points. Show that E has Lebesgue measure ζ.
(iii) Construct a set H ⊆ [0, 1] of first Baire category but of Lebesgue
measure 1. Points in H are ‘generic in the the sense of measure theory’
(almost all points in [0, 1] lie in H) but the points of [0, 1] \ H are ‘generic
in the the sense of topology’ (quasi-all points in [0, 1] lie [0, 1] \ H).
(iv) Construct a set P ⊆ R of first Baire category such that R \ P has
Lebesgue measure zero.
The third argument is that the act of seeking a Baire type proof may, by
itself, suggest a new ways of looking at your problem.
3 The Hausdorff metric
The Hausdorff metric measures the difference between compact sets.
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