Probability Measures in Product Spaces

Abstract

All of us have an intuitive notion of chance or randomness when we say, for example, that we are 90% sure that a particular event will occur. Two facts underlie this statement: one, that we are talking about a non-deterministic event and two, that we are quantifying, in a heuristic way, the attitude of our mind. Areas where the notion of chance manifests itself are numerous, some of which are risk assessment, reliability, physics of elementary particles and finance. Probability theory is concerned with developing a mathematical frame work to analyse chance phenomena. Event and probability will be basic notions of that theory. In giving mathematical content and shape to these concepts, we adopt the by now well accepted set theoretic model. In dealing with experiments whose outcomes depend on chance, one considers the set of all outcomes and would like to assign a probability value to every possible outcome. In the set theoretical model this is found not possible if the cardinal number of the set of all the outcomes is equal to or exceeds that of the continuum. This in practice turns out to be no infirmity since almost all outcomes of importance to the experiment can be assigned probabilities and the set theoretic model has proved to be highly fruitful. This model rests on the fundamental notions of sets, measures, measurable functions and integrals.