Excerpt from The Theory of Functions of a Real Variable and the Theory of Fourier's Series, by E. W Hobson

The foundations of the Integral Calculus, as based upon Riemann's definition of a definite integral, and its extensions, are discussed in Chapter V, where an account of the development of the subject from the point of View of Lebesgue's new definition of the definite Integral is also given. In later parts of the book I have introduced extensions of Lebesgue's definition, to the cases of improper integrals, taken over finite or infinite domains, regarded as the limits of sequences of Lebesgue integrals.

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