As was noted in Chapter 1, the framework of measure theory builds the proper mathematical foundation for probability theory, in a much more crucial way than perhaps it does for any other branch of mathematics. In Section 2.11, we elaborated the measure theoretic framework for probability theory. As mentioned earlier, Probability theory is the mathematics of random experiments and the mathematical set-up that is used to model a random experiment consists of a triplet \((\Omega , \mathcal{A}, P)\) , called a probability space, where \((\Omega , \mathcal{A})\) is a measurable space and P a probability measure on \(\mathcal{A}\) . As was noted in Section 2.11, \(\Omega \) represents the set of “possible outcomes” of a random experiment, \(\mathcal{A}\) is the class of “events” and P the probability assignment to various events. It was emphasized that except in a very special case, namely, the “discrete case”, the \(\sigma \) -field \(\mathcal{A}\) , consisting of “events”, may not include all subsets of \(\Omega \) . All of these and much more were discussed in Chapter 2. In this Chapter, we are going to see how the idea of measurable functions and the theory of integration discussed in the previous chapter, comes into play in probability theory. In course of this, we will be studying one of the central objects of interest in probability theory, namely, random variables and more generally, random vectors.