172 Modeling and Inverse Problems in the Presence of Uncertainty
Define the space C
B
(Ω
θ
) = {f : Ω
θ
→ R | f is bounded a nd continuous}
and let C
∗
B
(Ω
θ
) denote its topological dual.
Theorem 5.4.1 Assume the parameter space Ω
θ
with its metric
˜
d is sepa-
rable. Then the Prohorov metric metrizes the weak
∗
topology of the space
C
∗
B
(Ω
θ
) and the following are equivalent:
(i) ρ(P
M
, P ) → 0.
(ii)
Z
Ω
θ
f(θ)dP
M
(θ) →
Z
Ω
θ
f(θ)dP (θ) for boun ded, uniformly continuous
functions f .
(iii) P
M
(A) → P (A) for all Borel sets A ⊂ Ω
θ
with P (∂A) = 0.
We remark that the separability of the metric space (Ω
θ
,
˜
d) is not strictly nec-
essary. The so-called weak topology of probability measures (weak
∗
top ology)
and the topology induced by the Prohorov metric are equivalent provided ev-
ery ...