Baire Category, Probabilistic Constructions and Convolution Squares

then, automatically,
supp f ⊆ E ∩ suppFm.
Thus, by choosing an appropriate finite set A and setting H = A∪suppf,
we can ensure that (E,µ) ∈ Lβ,
(E,µ), (H,σ) < ǫ/4.
dβ
and we can find a finite collection of intervals I such that
I ⊇ E and
|I| α+1/n < 1
n .
I∈Im
I∈Im
We have shown that (setting dµ(t) = f(t)dt) (E,µ) ∈ Hn and all we need
to do is to show that, for appropriate choices of η and m we have
sup |
r∈Z
ˆ f(r) − ˆg(r)| < ǫ/4, f ∗ f − g ∗ g∞ < ǫ/4 and ωβ(f ∗ f − g ∗ g) < ǫ/4.
Without loss of generality we may suppose ǫ < 1, so simple calculations show
that it is sufficient to prove
sup |ˆg(r)−
r∈Z
ˆ Gm(r)| < ǫ/8, f ∗f −Gm∗Gm∞ < ǫ/8 and ωβ(f ∗f −g∗g) < ǫ/8.
Using (vii)m, we have
|ˆg(r) − ˆ
∞
Gm(r)| = ˆg(r)
− ˆg(r − j)
j=−∞
ˆ
Fm(j)
= ˆg(r − u)
ˆ
Fm(u) ≤ |ˆg(r − u)||
ˆ Fm(u)|
u=0
≤
|ˆg(r − u)|η ≤ η
u=0
∞
j=−∞
u=0
|ˆg(j)| < ǫ/8
for all r provided only that η is small enough. We now fix η once and for all
so that the inequality just stated holds and
η (1 + g∞) 2 + ωβ(g ∗ g) + 2) < ǫ/12
but leave m free.
We have now arrived at the central estimates of the proof which show
that
g ∗ g − Gm ∗ Gm∞ < ǫ/8 and ωβ(g ∗ g − Gm ∗ Gm) < ǫ/8,
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