Based on the notions of measure spaces and measurable maps, we introduce the integral of a measurable map with respect to a general measure. This generalises the Lebesgue integral that can be found in textbooks on calculus. Furthermore, the integral is a cornerstone in a systematic theory of probability that allows for the definition and investigation of expected values and higher moments of random variables.

## Keywords

Normal Representation Monotone Convergence Monotone Convergence Theorem Nonnegative Measurable Function Lebesgue Integrable Function
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© Springer-Verlag London Limited 2008