The intent of this short chapter is to indicate how the previous theory may be extended in an obvious way to include the integration of complex-valued functions with respect to a measure (or signed measure) μ on a measurable space (X, S). The primary purpose of this is to discuss Fourier and related transforms which are important in a wide variety of contexts – and in particular the Chapter 12 discussion of characteristic functions of random variables which provide a standard and useful tool in summarizing their probabilistic properties.

Some standard inversion theorems will be proved here to help avoid overload of the Chapter 12 material. However, methods of this chapter also apply to other diverse applications e.g. to Laplace and related transforms used in fields such as physics as well as in probabilistic areas such as stochastic modeling, and may be useful for reference.

Finally it might be emphasized (as noted later) that the integrals considered here involve complex functions as integrands and as for the preceding development, form a “Lebesgue-style” theory. This is in contrast to what is termed “complex variable” methodology, which is a “Riemann-style” theory in which integrals are considered with respect to a complex variable z along some curve in the complex plane. The latter methods – not considered here – can be especially useful in providing means for evaluation of integrals such as characteristic functions which may resist simple real variable techniques.