In this chapter we tackle the first major step in measure theory. To recapitulate, we had seen that to pursue with Lebesgue’s idea of integration, a fundamental requirement is to be able to assign a notion of length to arbitrary subsets of the real line. Similarly, in the infinite coin tossing experiment, we have the sample space \(\Omega \) , consisting of all infinite sequences of H and T, and our aim is to assign probabilities to arbitrary subsets of \(\Omega \) , starting from the naturally assigned probabilities to some special subsets.