Construction and stochastic applications of measure spaces in higher-order logic

A rich formalization of measure and probability theory is a prerequisite to analyzing probabilistic program behaviour in interactive theorem provers. When using probability theory it is important to have the necessary tools (definitions and theorems) to construct measure spaces with the desired properties. This thesis presents the formalization of a rich set of constructions. We start with discrete measures, distributions, densities and products. Then, we introduce the Lebesgue measure and products of probability spaces with an infinite index. The latter is used to construct the stochastic processes of discrete-time Markov chains on discrete state spaces. For applications like randomized algorithms discrete probability spaces are enough. Here single elements have nonzero measures assigned. More advanced constructions are needed when we measure infinite traces, since sets of traces are not discrete. Important models for such trace spaces are discrete-time Markov chains. Here a trace is a sequence of states and the probability which state to choose next only depends on the previous state. We construct the trace measure of a Markov chain, based on the transition probabilities between states. With the formalization of Markov chains we verify probabilistic model checking, anonymity in the Crowds protocol, and the probability to allocate a free address in the ZeroConf protocol. This development is done in the interactive theorem prover Isabelle/HOL.