ABSTRACT
This chapter presents a rigorous treatment of probability theory from a mathematical perspective. It illustrates the usefulness of the Sierpinski class in defining the probability measure extension. The chapter provides the general definition of a random object, and explains the two important concepts, namely measurability and probability spaces. It discusses the uniqueness of probability measures via uniqueness in characteristic functions. The distribution theory based on rudimentary quantities such as densities and cumulative distribution function, which are related to probability measures on the measurable space. Expectation provides a useful tool to help readers to study random objects and their distribution. Since expectation of a random variable is by definition an integral of some measurable function, it inherits all the properties from general integration theory, such as linearity and monotonicity. The chapter discusses conditioning given the value of a random variable via rigorous arguments.
