By Prakash Panangaden
Labelled Markov strategies are probabilistic types of labelled transition platforms with non-stop country areas. This ebook covers uncomplicated likelihood and degree idea on non-stop kingdom areas after which develops the idea of LMPs. the most subject matters lined are bisimulation, the logical characterization of bisimulation, metrics and approximation idea. An strange characteristic of the ebook is the relationship made with specific and area theoretic suggestions.
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Extra info for Labelled Markov Processes
Thus we have a countable family of open rectangles which generate the topology of Y and hence the Borel algebra BY . These are all in the σ-algebra B1 ⊗ B2 . Now any open set is the countable union of these special rectangular open labelled May 11, 2009 11:38 WSPC/Book Trim Size for 9in x 6in Measure Theory labelled 21 sets and thus any open set in the product topology of Y is in B1 ⊗ B2 . Thus we have that BY ⊆ B1 ⊗B2 . Now consider the identity function on Y viewed → (Y, B1 ⊗ B2 ). e. iY : (Y, BY ) − This function is the one induced by universality of the product given the projections onto the components.
Labelled May 11, 2009 11:38 WSPC/Book Trim Size for 9in x 6in Measure Theory labelled 29 We need to verify that this is independent of the chosen A and B. Suppose that we have another pair A ⊆ Z ⊆ B with µ(B − A ) = 0. Now since B ⊇ Z ⊇ A and B ⊇ Z ⊇ A , we have µ(B ) ≥ µ(A) = µ(B) and µ(B) ≥ µ(A ) = µ(B ). Thus we have µ(A) = µ(A ) = µ(B) = µ(B ) and µ is well deﬁned. Finally we need to verify countable additivity of µ . Suppose that we have a pairwise disjoint family of sets Zn from Σ . Then the associated sets An are also pairwise disjoint, thus we have µ (∪n Zn ) = µ(∪n An ) = µ(An ) = n µ (Zn ).
The proofs are elementary and are left to the exercises. The last is a triviality. The next proposition is the ﬁrst step towards establishing linearity of the integral. 5 Let s, t be simple integrable functions. Then (s + t)µ = sµ + tµ. Proof. Let A be a measurable set, s any simple integrable function and deﬁne φ(A) = A sµ. Then φ is a measure. To see this recall that any simple function can be written in the form of a ﬁnite sum n a i χA i s= i=1 where for deﬁniteness we assume that the ai are distinct.