Baire Category, Probabilistic Constructions and Convolution Squares

Exercise 5.8. We work in R 2 . Let Q to be the collection of all closed subsets
Q of the disc centre 0 radius 2 with the following properties.
(i) Q is the union of line segments of length at least 1 joining points on
the boundary of the disc.
(ii) We can find a line segment of the type described in (i) in every direction.
Show that, for an appropriate complete metric space, quasi-all elements of Q
have Lebesgue measure 0.
6 Measures
For the remainder of the lectures we shall be interested in measures, their
supports and their Fourier transforms. This section is not intended to be
complete, but merely intended to establish notation and to jog the reader’s
memory. Later, I shall use results on measures which include several not
mentioned here
We shall consider the space M(T) of Borel measures µ on T. From our
point of view, the two key properties of Borel measures are that, if µ ∈ M(T),
then
f dµ is defined for all f ∈ C(T) and, that, if µ, τ ∈ M(T) satisfy
T
f dµ = f dτ
T
for all f ∈ C(T), then µ = τ.
We recall that M(T) has a natural norm
µ = sup f dµ
: f ∈ C(T), f∞
≤ 1 .
T
Standard theorems tell us that the unit ball for this norm is weakly compact,
that is to say, that, given µn ∈ M(T) with µn ≤ 1, we can find n(j) → ∞
and µ ∈ M(T) such that
f dµn(j) → f dµ
T
for all f ∈ C(T).
We say that a measure µ is positive if
f dµ is real and positive for all
T
f ∈ C(T) with f(t) real and positive for all t ∈ T. A probability measure is
a positive measure of norm 1.
Exercise 6.1. Show that the set of probability measures is closed under the
standard norm. Show that it is weakly compact.
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