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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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