59.19. CHARACTERISTIC FUNCTIONS, PROKHOROV THEOREM 1927

=∫[|x j|≥ 2

u ]1dµn (x)

= µn

([x :∣∣x j∣∣≥ 2

u

]).

If ε > 0 is given, there exists r > 0 such that if u≤ r,

1u

∫ u

−u(1−ψ (te j))dt < ε/p

for all j = 1, · · · , p and so, by the dominated convergence theorem, the same is true withφ µn

in place of ψ provided n is large enough, say n≥ N (u). Thus, from 59.19.46, if u≤ r,and n≥ N (u),

µn

([x :∣∣x j∣∣≥ 2

u

])< ε/p

for all j ∈ {1, · · · , p}. It follows that for u≤ r and n≥ N (u) ,

µn

([x : ||x||

∞≥ 2

u

])< ε.

because [x : ||x||

∞≥ 2

u

]⊆ ∪p

j=1

[x :∣∣x j∣∣≥ 2

u

]This proves the lemma because there are only finitely many measures, µn for n < N (u) andthe compact set can be enlarged finitely many times to obtain a single compact set, Kε suchthat for all n,µn ([x /∈ Kε ])< ε.

As before, there are simple modifications of Lemmas 59.18.3 and 59.18.4. The first ofthese is as follows.

Lemma 59.19.3 If φ µn(t)→ φ µ (t) for all t, then whenever ψ ∈S, the Schwartz class,

µn (ψ)≡∫Rp

ψ (y)dµn (y)→∫Rp

ψ (y)dµ (y)≡ µ (ψ)

as n→ ∞.

Proof: By definition,

φ µ (y)≡∫Rp

eiy·xdµ (x) .

Also remember the inverse Fourier transform. Letting ψ ∈S, the Schwartz class,

F−1 (µ)(ψ) ≡ µ(F−1

ψ)≡∫Rp

F−1ψdµ

=1

(2π)p/2

∫Rp

∫Rp

eiy·xψ (x)dxdµ (y)

=1

(2π)p/2

∫Rp

ψ (x)∫Rp

eiy·xdµ (y)dx

=1

(2π)p/2

∫Rp

ψ (x)φ µ (x)dx