Probability and Stochastic Processes

Chapter 2Random Variables

All the definitions with sets presented in Chapter 1 are consistent, but if we wish to calculate and compute numerical values related to abstract spaces we need to standardize the spaces. The first step is to give the following definition.

Definition 2.1 (Measurable function (m.f.))

Let , be two measurable spaces. Let be a function. f is called a measurable function if and only if for any set we have The inverse function is a set function defined in terms of the pre-image. Explicitly, for a given set ,

Note: This definition makes it possible to extend probability measures to other spaces. For instance, let f be a measurable function and assume that there exists a probability measure on the first space Then we can construct a probability measure on the ...