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Using Lemma
6.6
(iii) once more, we have
ʘ
1
,
ʘ
2
such that
ʘ
1
,
i
ʘ
2
,
ʘ
1
=
ʘ
2
,
ʔ
1
⃒
ʔ
i
⃒
ʘ
1
+
≥
2
ʘ
2
. Continuing this process we have that
thus in combination
ʘ
=
ʘ
0
+
ʘ
1
+
i
ʔ
j
⃒
ʘ
i
+
1
,
ʘ
i
+
1
ʔ
i
⃒
ʘ
i
,
ʘ
=
ʘ
j
+
j
≥
i
+
1
j
=
0
|
j
≥
i
+
1
ʔ
j
|≥|
ʘ
i
+
1
|
for all
i
0.
But since
i
=
0
ʔ
i
is a subdistribution, we know that the tail sum
j
≥
i
+
1
ʔ
j
converges to
ʵ
when
i
approaches
≥
0. Lemma
6.6
(i) ensures that
for all
i
≥
, and therefore that lim
i
ₒ∞
ʘ
i
∞
=
ʵ
. Thus
=
i
=
0
ʘ
i
.
by taking that limit we conclude that
ʘ
With Theorem
6.5
, the relation
⃒ ↆ
D
sub
(
S
)
×
D
sub
(
S
) can be obtained as the
lifting of a relation
⃒
S
from
S
to
D
sub
(
S
), which is defined by writing
s
⃒
S
ʘ
just when
s
⃒
ʘ
.
⃒
S
)
†
Proposition 6.3
(
=
(
⃒
)
.
⃒
S
)
†
ʘ
implies
ʔ
⃒
Proof
ʘ
is a simple application of Part (i) of Th
e
-
orem
6.5
. For the other direction, suppose
ʔ
That
ʔ
(
=
s
∈
ʔ
ʔ
(
s
)
⃒
·
ʘ
. Given that
ʔ
s
,
P
art (ii) of the same theorem enables us to decompose
ʘ
into
s
∈
ʔ
ʔ
(
s
)
·
ʘ
s
, where
s
⃒
ʘ
s
for each
s
in
ʔ
. But the latter actually means that
s
⃒
S
ʘ
s
, and so by
⃒
S
)
†
ʘ
.
definition this implies
ʔ
(
is convex because of its being a lifting.
We proceed with the important properties of reflexivity and transitivity of weak
derivations. First note that reflexivity is straightforward; in Definition
6.4
it suffices
to take
ʔ
0
It is immediate that the relation
⃒
to be the empty subdistribution
ʵ
.
Theorem 6.6 (Transitivity of
⃒
)
If ʔ
⃒
ʘ and ʘ
⃒
ʛ then ʔ
⃒
ʛ.
ʘ
means that some
ʔ
k
,
ʔ
k
,
ʔ
k
Proof
By definition
ʔ
⃒
exist for all
k
≥
0
such that
∞
˄
−ₒ
ʔ
k
+
ʔ
k
,
ʔ
k
ʔ
k
.
ʔ
=
ʔ
0
,
ʔ
k
=
ʔ
k
+
1
,
ʘ
=
(6.7)
k
=
0
Since
ʘ
=
k
=
0
ʔ
k
and
ʘ
⃒
ʛ
, by Theorem
6.5
(ii) there are
ʛ
k
for
k
≥
0
=
k
=
0
ʛ
k
and
ʔ
k
⃒
such that
ʛ
ʛ
k
for all
k
≥
0.
0, we know that
ʔ
k
⃒
ʛ
k
gives us some
ʔ
kl
,
ʔ
kl
,
ʔ
kl
Now for each
k
≥
for
l
≥
0 such that
˄
−ₒ
ʔ
k
=
ʔ
kl
+
ʔ
kl
.
(6.8)
ʔ
kl
,
ʔ
kl
ʔ
k
0
,
ʔ
kl
=
ʔ
k
,
l
+
1
ʛ
k
=
l
≥
0
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