On Strongly Equivalent Nonrandomized Transition Probabilities
Eugene A. Feinberg
1
and Alexey B. Piunovskiy
2
Abstract
Dvoretzky, Wald and Wolfowitz proved in 1951 the existence of equivalent and strongly equivalent
mappings for a given transition probability when the number of nonatomic measures is finite and the
decision set is finite. This paper introduces a notion of strongly equivalent transition probabilities with
respect to a finite collection of functions. This notion contains the notions of equivalent and strongly
equivalent transition probabilities as particular cases. This paper shows that a strongly equivalent map
ping with respect to a finite collection of functions exists for a finite number of nonatomic distributions
and finite decision set. It also provides a condition when this is true for a countable decision set. Ac
cording to a recent example by Loeb and Sun, a strongly equivalent mapping may not exist under these
conditions when the decision set is uncountable. This paper also provides two additional counterexam
ples and shows that strongly equivalent mappings exist for homogeneous transition probabilities.
1
Introduction
Dvoretzky, Wald and Wolfowits [2, 3] studied the following problem. Consider a standard Borel
space
X
, i.e.
X
is a onetoone measurable image of a Borel subset of a Polish space, and consider
another standard Borel space
A
. Sometimes
X
is called a state space (or set) and
A
is called
a decision space (or set). Let a finite number of nonatomic probability measures
μ
1
, . . . , μ
K
on
X
be given.
We recall that a measure
μ
on a standard Borel space
X
is called nonatomic if
μ
(
{
x
}
) = 0
for all
x
∈
X
. We follow the terminology that a finite set is countable. Of course, if
A
is countable then
R
A
f
(
a
)
ν
(
da
) =
∑
a
∈
A
f
(
a
)
ν
(
a
)
for any measure
ν
and for any function
f
on
A
.
Denote by
π
(
da

x
)
a regular transition probability from
X
to
A
. Since we consider only regular
transition probabilities and only measurable functions, we sometimes omit the terms “regular” and
“measurable.” The following two facts were proved in [3].
Proposition 1.
([3, Theorem 3.1]) If
A
is a finite set then for any finite number
K
, for any mea
surable functions
r
k
(
x, a
)
, k
= 1
, . . . , K,
and for any transition probability
π
from
X
to
A
, there
exists a measurable mapping
ϕ
of
X
to
A
such that
Z
X
Z
A
r
k
(
x, a
)
π
(
da

x
)
μ
k
(
dx
) =
Z
X
r
k
(
x, ϕ
(
x
))
μ
k
(
dx
)
,
k
= 1
, . . . , K.
1
Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794
3600, USA.
2
Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK.
1
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Proposition 2.
([3, Theorem 3.2]) If
A
is a finite set then for any transition probability
π
from
X
to
A
there exists a measurable mapping
ϕ
:
X
→
A
such that
Z
X
π
(
a

x
)
μ
k
(
dx
) =
μ
k
(
{
x
:
ϕ
(
x
) =
a
}
)
,
a
∈
A
, k
= 1
, . . . , K.
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 Fall '11
 EugeneA.Feinberg
 Probability space, Lebesgue integration, transition probability, Borel measure

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