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(i) It only works for integration of functions defined in Rn.

(ii) It can only deal will integration of functions with respect to the Lebesgue measure, i.e. the usual measure, on Rn.

(iii) Unbounded functions have to be treated separately.

(iv) The theory lacks certain convergence properties. For example, the pointwise limit of a uniformly bounded sequence of Riemann integrable functions may fail to be Riemann integrable.

(v) A function with a `large' set of discontinuity, i.e. with non-zero Lebesgue measure, does not have a Riemann integral.

In the early years of this century, Lebesgue and Borel, amongst others, laid the foundation of a new theory of integration. With its further development, the new theory, the so-called Lebesgue integration, has become the basis of measure theory and functional analysis. A special case of the Lebesgue integral, the so-called Lebesgue-Stieltjes integral, has also played a fundamental role in probability theory. The underlying basis of the Lebesgue theory is in sharp contrast to that of the Riemann theory. Whereas, in the theory of Riemann integration, the domain of the function is partitioned and the integral of the function is approximated by the lower and upper Darboux sums induced by the partition, in the theory of Lebesgue integration, the range of the function is partitioned to produce simple functions which approximate the function, and the integral is defined as the limit of the integrals of these simple functions. The latter framework makes it possible to define the integral of measurable functions on abstract measurable spaces, in particular on topological spaces equipped with Borel measures. Lebesgue integration also enjoys very general convergence properties, giving rise to the complete Lp-spaces. Moreover, when the Riemann integral of a function exists, so does its Lebesgue integral and the two values coincide, i.e. Lebesgue integration includes Riemann integration. Nevertheless, despite these desired features, Lebesgue integration is quite involved and much less constructive than Riemann integration. Consequently, Riemann integration remains the preferred theory wherever it is adequate in practice, in particular in advanced calculus and in the theory of differential equations.

A number of theories have been developed to generalise the Riemann integral while trying to retain its constructive quality. The most well-known and successful is of course the Riemann-Stieltjes integral. In more recent times, E. J. McShane [22] has developed a Riemann-type integral, which includes for example the Lebesgue-Stieltjes integral, but it unfortunately falls short of the constructive features of the Riemann integral.

A new idea in measure theory on second countable locally compact Hausdorff spaces was presented in [9]. It was shown that the set of normalised Borel measures on such a space can be embedded into the maximal elements of the probabilistic power domain of its upper space. The image of the embedding consists of all normalised valuations on the