ABSTRACT
This chapter presents some measure-theoretic foundations of probability theory and random variables. This then leads us into the main tools and formulas for computing expectations and conditional expectations of random variables (more importantly, continuous random variables) under different probability measures. The main formulas that are provided are used for the understanding and quantitative modelling of continuous-time stochastic financial models. Simple random variables are particularly useful and provide an alternative equivalent way of defining the Lebesgue integral. There are peculiar Borel sets that are uncountable but yet have Lebesgue measure zero. The integration theory for Borel functions of a single variable (and random variables defined as functions of a single random variable) extends into the general multidimensional case. The chapter discusses how distributions and expectations are formulated for multiple random variables (i.e., random vectors) by first considering a pair of random variables.
