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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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