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6.3.3
Properties of Weak Transitions
Here, we develop some properties of the weak move relation
that will be im-
portant later on. We wish to use weak derivation as much as possible in the same
way as the lifted action relations
ʱ
−ₒ
, and therefore, we start with showing that
ʱ
−ₒ
enjoys two of the most crucial properties of
: linearity of Definition 6.2 and the
decomposition property of Proposition 6.2 . To this end, we first establish that weak
derivations do not increase the mass of distributions and are preserved under scaling.
Lemma 6.6
For any subdistributions ʔ, ʘ, ʓ , ʛ, ʠ we have
(i) If ʔ
ʘ then
|
ʔ
|≥|
ʘ
|
.
(ii) If ʔ
ʘ and p
∈ R 0 such that
|
p
·
ʔ
|≤
1 , then p
·
ʔ
p
·
ʘ.
ʠ ʓ
ʠ ʛ with ʓ
ʠ ʓ
ʠ ʛ .
(iii) If ʓ
+
ʛ
ʠ then ʠ
=
+
and 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
=
k = 0
A simple inductive proof shows that
k i |
ʔ i |+
ʔ k |
|
ʔ
|=|
for any i
0.
(6.5)
{ k i |
ʔ k |} i = 0
The sequence
is nondecreasing and by ( 6.5 ) each element of the
sequence is not greater than
|
ʔ
|
. Therefore, the limit of this sequence is bounded by
|
ʔ
|
. That is,
k i | ʔ k |=| ʘ | .
| ʔ |≥
lim
i ₒ∞
Now suppose p
∈ R 0 such that
|
p
·
ʔ
|≤
1. From Remark 6.2 (i) it follows that
˄
−ₒ
ʔ k +
ʔ k ,
ʔ k
ʔ k .
p
·
ʔ
=
p
·
ʔ 0 ,
p
·
ʔ k =
p
·
p
·
p
·
p
·
ʔ k + 1 ,
p
·
ʘ
=
p
·
k
Hence, Definition 6.4 yields p
·
ʔ
p
·
ʘ .
Next suppose ʓ
+
ʛ
ʠ . By Definition 6.4 there are subdistributions ʠ k ,
, ʠ k
ʠ k
for k
∈ N
such that
˄
−ₒ
ʠ k
ʠ k ,
ʠ k
ʠ k .
ʓ
+
ʛ
=
ʠ 0 ,
ʠ k =
+
ʠ k + 1 ,
ʠ
=
k
For any s
S , define
ʓ 0
min ( ʓ ( s ), ʠ 0
( s ):
=
( s ))
ʓ 0 ( s ):
ʓ 0
=
ʓ ( s )
( s )
(6.6)
ʛ 0 ( s ):
min ( ʛ ( s ), ʠ 0 ( s ))
=
ʛ 0
ʛ 0 ( s ),
( s ):
= ʛ ( s )
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