Probabilistic Methods in Concurrency Lecture 5 Basics of Measure Theory and Probability Theory Probabilistic Automata Catuscia Palamidessi catuscia@lix.polytechnique.fr.

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Probabilistic Methods in Concurrency Lecture 5 Basics of Measure Theory and Probability Theory Probabilistic Automata Catuscia Palamidessi catuscia@lix.polytechnique.fr www.lix.polytechnique.fr/~catuscia Page of the course: www.lix.polytechnique.fr/~catuscia/teaching/Pisa/

Basics of Measure Theory and Continuous Probability Theory Continuous probabilistic spaces: the need for Measure Theory Some introductory examples: Infinite sequence of coin tossing, Intersection, complementation, union and countable union Why countable Concept of cone Measurable space, s-field Base, s-field generated by a base Examples Probability measure Monotonicity and continuity Prakash Panangaden, Stochastic techniques in Concurrency. Lect notes. Section 2.1. and 2.4 (till page 16) Pisa, 2 July 2004 Prob methods in Concurrency

Probabilistic Automata Nondeterministic choice and probabilistic choice Definition of probabilistic automata Concept of adversary Concept of execution The measurable space and the probability measure associated to the executions Some examples Roberto Segala. Modeling and Verification of Randomized Distributed Real Time Systems . PhD thesis, Laboratory for Computer Science, Massachusetts Institute of Technology, June 1995. Available as Technical Report MIT/LCS/TR-676. Roberto Segala and Nancy Lynch. Probabilistic simulations for probabilistic processes. Nordic Journal of Computing, 2(2):250--273, 1995. An extended abstract appeared in the Proceedings of CONCUR '94, LNCS 836: 22--25. Pisa, 2 July 2004 Prob methods in Concurrency