Contents Preface 5 Chapter 1. Probability, measure and integration 7 1.1. Probability spaces, measures and σ-algebras 7 1.2. Random variables and their distribution 18 1.3. Integration and the (mathematical) expectation 30 1.4. Independence and product measures 54 Chapter 2. Asymptotics: the law of large numbers 71 2.1. Weak laws of large numbers 71 2.2. The Borel-Cantelli lemmas 77 2.3. Strong...

The paper addresses the problem of computing maximal conditional expected accumulated rewards until reaching a target state (briefly called maximal conditional expectations) in finite-state Markov decision processes where the condition is given as a reachability constraint. Conditional expectations of this type can, e.g., stand for the maximal expected termination time of probabilistic programs...

converge to ET (x) in A for any x ∈ A, as n → ∞, here ET is an projection of A to the fixed point subspace of T . Note that when T is a completely positive mapping then ET is a conditional expectation. We provide an example of uniquely ergodic dynamical system relative its fixed point subspace, for which ET is not a conditional expectation. It is also constructed a uniquely ergodic entangled Ma...

We extend a result of Dubins (1975) from bounded to unbounded random variables. Dubins (1975) showed that a finitely additive expectation over the collection of bounded random variables can be written as an integral of conditional expectations (disintegrability) if and only if the marginal expectation is always within the smallest closed interval containing the conditional expectations (conglom...

Let A be a C∗-algebra and E : A → A a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, E(x)∗E(x) ≤ E(x∗x), implies that ‖E(x)‖ ≤ ‖E(x∗x)‖ . In this note we show that E is homomorphic (in the sense that E(xy) = E(x)E(y) for every x, y in A) if and only if ‖E(x)‖ = ‖E(x∗x)‖ , for every x in A. We also prove that a homomorphic conditional expectation on a commu...