Baire Category, Probabilistic Constructions and Convolution Squares

Theorem 2.2. Let (X,d) be a non-empty complete metric space. Suppose
that Pj is a property such that:-
(i) The property of being Pj is stable in the sense that, given x ∈ X
which has property Pj, we can find an ǫ > 0 such that whenever d(x,y) < ǫ
the point y has the property Pj.
(ii) The property of not being Pj is unstable in the sense that, given
x ∈ X and ǫ > 0, we can find a y ∈ X with d(x,y) < ǫ which does not have
the property Pj.
Then there is an x0 ∈ X which has all of the of the properties P1, P2,
....
Proof of equivalence of Theorems 2.1 and 2.2. Let x have the property Pj if
and only if x /∈ Ej.
It is unlikely that anyone who reads these notes is unfamiliar with Theorem
2.1 or its proof, but I include a proof for completeness. We shall prove
a slightly stronger version of Baire’s theorem.
Theorem 2.3. Let (X,d) be a complete metric space. If E1, E2, ...are
closed sets with empty interiors, then X \ ∞
j=1 Ej is dense in X.
Proof. Suppose that x0 ∈ X and δ0 > 0. We shall show that there exists a
y ∈ B(x0,δ0) such that y /∈ ∞ j=1 Ej.
To this end, we perform the following inductive construction. Given
xn−1 ∈ X and δn > 0, we can find xn ∈ X such that xn ∈ B(xn−1,δn/4), but
xn /∈ En. (For, if not, we would have B(xn−1,δn−1/4) ⊆ En and En would
have a non-empty interior.) Since En is closed and xn /∈ En we can now find
a with δn−1/2 > δn > 0 such that B(xn,δn) En = ∅.
Now observe that
δm ≤ 2 −1 δm−1 ≤ 2 −2 δm−2 ≤ ...2 n−m δn
for all m ≥ n ≥ 0. It follows that, if r ≥ s
s−1
d(xr,xs) ≤
j=r
d(xj+1,xj) ≤
s−1
δr/4 ≤ 4 −1 s−1
δ0
j=r
j=r
2 −r ≤ 2 −r−1 δ0.
Thus the xr form a Cauchy sequence and converges in (X,d) to some point
y.
The same kind of calculation as in the last paragraph gives
s−1
d(xr,xs) ≤
j=r
d(xj+1,xj) ≤
s−1
δr/4 ≤ 4 −1 δr
j=r
4
s−r−1
j=0
2 −r ≤ δr/2,