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Modeling and Inverse Problems in the Presence of Uncertainty
book

Modeling and Inverse Problems in the Presence of Uncertainty

by H. T. Banks, Shuhua Hu, W. Clayton Thompson
April 2014
Intermediate to advanced content levelIntermediate to advanced
405 pages
13h
English
Chapman and Hall/CRC
Content preview from Modeling and Inverse Problems in the Presence of Uncertainty
172 Modeling and Inverse Problems in the Presence of Uncertainty
Define the space C
B
(Ω
θ
) = {f :
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
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
θ
f(θ)dP
M
(θ)
Z
θ
f(θ)dP (θ) for boun ded, uniformly continuous
functions f .
(iii) P
M
(A) P (A) for all Borel sets A
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 ...
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Publisher Resources

ISBN: 9781482206432