Baire Category, Probabilistic Constructions and Convolution Squares

Proof. It is sufficient to show that, if En ∈ E and dE(En,En+1) ≤ 2 −n−1 for
all n ≥ 1, then En converges in the Hausdorff metric.
To this end, let E be the set of e ∈ X such that there exist en ∈ En with
d(en,e)) → 0 as n → ∞. We observe that, if e ∈ E then, given any m, we
can find an n ≥ m + 1 such that d(e,en) < 2 −m . Since
n−1
dE(Em,En) ≤
j=m
n−1
dE(Ej,Ej+1) ≤ 2 −(j+1) < 2 −m
we can find xm ∈ Em such that d(xm,en) < 2 −m and so d(xm,e) < 2 −m+1 .
Thus
E ⊇ {x : d(x,xm) < 2 −m+1 for some x ∈ Em.
We next show that E is compact. Suppose that y(j) ∈ E for j ≥ 1.
We construct infinite subsets An of N as follows. Set A0 = N. If Am−1
has been defined we obtain Am as follows. Since E is covered by open balls
B(x, 2 −m+1 ) with x ∈ Em and E is compact, E is covered by a finite set
of such balls and one of those balls B(xm, 2 −m+1 ) must contain an infinite
subset of Am. We observe that d(xm,xm+1) < 2 −m so the xm converge to
some y ∈ E. Choose n(j) ∈ Aj so that n(j) → ∞. Then d(yn(j),y) → 0 as
j → ∞. Thus E is compact.
The second paragraph of the proof shows that
j=m
sup inf
f∈En
e∈E d(e,f) ≤ 2−n+1 .
If xn ∈ En then we can find xj ∈ Ej such that d(xj,xj+1) < 2 −j+1 . Since the
xj are Cauchy, they converge to some x. We have x ∈ E and
so
d(x,xn) ≤
sup
e∈E
inf
∞
d(xj,xj+1) ≤ 2 −n+2
j=n
f∈En
d(e,f) ≤ 2 −n+2 .
Thus dE(En,E) → 0 as n → ∞.
Exercise 3.5. (We use the notation of Definition 3.1.) Show that, if (E,dE)
is complete, then (X,d) is.
Exercise 3.6. In these notes, we are not interested in metric spaces in general
but in spaces like [0, 1] n , T n and R n with the usual Euclidean metric. We
can then give a simpler proof of the completeness of the Hausdorff metric.
8