whenever n is large enough by Lemma 58.19.3 because ψη ∈S. This establishes the
conclusion of the lemma in the case where ψ is also infinitely differentiable. To
consider the general case, let ψ only be uniformly continuous and let ψ_{k} = ψ ∗ ϕ_{k}
where ϕ_{k} is a mollifier whose support is in
(− (1∕k),(1∕k))
^{p}. Then ψ_{k} converges
uniformly to ψ and so the desired conclusion follows for ψ after a routine estimate.
■
The next theorem is really important. It gives the existence of a measure based on an
assumption that a set of measures is tight. The next theorem is Prokhorov’s theorem
about a tight set of measures. Recall that Λ is tight means that for every ε > 0 there
exists K compact such that μ
(KC )
< ε for all μ ∈ Λ.
Theorem 58.19.5Let Λ =
{μn}
_{n=1}^{∞}be a sequence of probability measures defined onthe Borel sets of ℝ^{p}. If Λ is tight then there exists a probability measure, λ and asubsequence of
{μn}
_{n=1}^{∞}, still denoted by
{μn}
_{n=1}^{∞}such that whenever ϕ is acontinuous bounded complex valued function defined on E,
∫ ∫
lim ϕdμn = ϕdλ.
n→∞
Proof: By tightness, there exists an increasing sequence of compact sets,
{Kn }
such
that
1
μ(Kn ) > 1− n
for all μ ∈ Λ. Now letting μ ∈ Λ and ϕ ∈ C
(Kn )
such that
||ϕ||
_{∞}≤ 1, it follows
||∫ ||
|| ϕdμ|| ≤ μ(Kn ) ≤ 1
Kn
and so the restrictions of the measures of Λ to K_{n} are contained in the unit ball of
C
(Kn)
^{′}. Recall from the Riesz representation theorem, the dual space of C
(Kn )
is a
space of complex Borel measures. Theorem 15.5.5 on Page 1327 implies the unit ball of
C
(Kn)
^{′} is weak ∗ sequentially compact. This follows from the observation that
C
(Kn)
is separable which follows easily from the Weierstrass approximation
theorem. Thus the unit ball in C
(Kn)
^{′} is actually metrizable by Theorem 15.5.5 on
Page 1327. Therefore, there exists a subsequence of Λ,
{μ1k}
such that their
restrictions to K_{1} converge weak ∗ to a measure, λ_{1}∈ C
(K1)
^{′}. That is, for every
ϕ ∈ C
(K1 )
,
∫ ∫
lim ϕd μ = ϕd λ
k→∞ K1 1k K1 1
By the same reasoning, there exists a further subsequence
{μ2k}
such that the
restrictions of these measures to K_{2} converge weak ∗ to a measure λ_{2}∈ C
(K2 )
^{′} etc.
Continuing this way,
′
μ11,μ12,μ13,⋅⋅⋅ → Weak ∗ in C (K1)′
μ21,μ22,μ23,⋅⋅⋅ → Weak ∗ in C (K2)′
μ31,μ32,μ33,⋅⋅⋅ → Weak ∗ in C (K3)
...
Here the j^{th} sequence is a subsequence of the
(j − 1)
^{th}. Let λ_{n} denote the measure in
C
(Kn)
^{′} to which the sequence
{μnk}
_{k=1}^{∞} converges weak ∗. Let
{μn}
≡
{μnn}
, the
diagonal sequence. Thus this sequence is ultimately a subsequence of every one of
the above sequences and so μ_{n} converges weak ∗ in C
(Km )
^{′} to λ_{m} for each
m.
Claim:For p > n, the restriction of λ_{p} to the Borel sets of K_{n} equals λ_{n}.
Proof of claim: Let H be a compact subset of K_{n}. Then there are sets, V_{l} open in
K_{n} which are decreasing and whose intersection equals H. This follows because this is a
metric space. Then let H ≺ ϕ_{l}≺ V_{l}. It follows
∫ ∫
λ (V ) ≥ ϕ dλ = lim ϕdμ
n l Kn l n k→ ∞ Kn l k
∫ ∫
= lk→im∞ K ϕldμk = K ϕldλp ≥ λp (H ).
p p
Now considering the ends of this inequality, let l →∞ and pass to the limit to
conclude
λ (H ) ≥ λ (H ).
n p
Similarly,
∫ ∫
λ (H ) ≤ ϕ dλ = lim ϕdμ
n Kn l n k→∞ Kn l k
∫ ∫
= lk→im∞ K ϕldμk = K ϕldλp ≤ λp (Vl).
p p
Then passing to the limit as l →∞, it follows
λ (H ) ≤ λ (H ).
n p
Thus the restriction of λ_{p},λ_{p}|_{Kn} to the compact sets of K_{n} equals λ_{n}. Then by inner
regularity it follows the two measures, λ_{p}|_{Kn}, and λ_{n} are equal on all Borel sets of K_{n}.
Recall that for finite measures on the Borel sets of separable metric spaces, regularity is
obtained for free.
It is fairly routine to exploit regularity of the measures to verify that λ_{m}
(F)
≥ 0 for
all F a Borel subset of K_{m}. (Whenever ϕ ≥ 0,∫_{Km}ϕdλ_{m}≥ 0 because ∫_{Km}ϕdμ_{k}≥ 0.
Now you can approximate X_{F} with a suitable nonnegative ϕ using regularity of the
measure.) Also, letting ϕ ≡ 1,
1 ≥ λm (Km ) ≥ 1−-1. (58.19.47)
m
(58.19.47)
Define for F a Borel set,
λ(F ) ≡ nli→m∞ λn (F ∩ Kn ).
The limit exists because the sequence on the right is increasing due to the above
observation that λ_{n} = λ_{m} on the Borel subsets of K_{m} whenever n > m. Thus for
n > m