In the very first chapter we have seen the usefulness of integration of simple functions on a boolean space. In many problems of probability and statistics random variables which are not necessarily simple, do arise and it is necessary to define the ‘average value’ or ‘expectation’ of such quantities. This can be achieved by extending the notion of integral further. It is also worth noting that mechanical concepts like centre of mass, moment of inertia, work, etc., can be formulated precisely in terms of integrals. However, in the initial stages of its development, the theory of integrals received its first push from the hands of the French mathematician H. Lebesgue on account of many new problems that arose in analysing the convergence properties of Fourier series. In the present chapter we shall introduce the idea of integral with respect to a measure on any borel space and investigate its basic properties.
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