Lebesgue integration

Wavelet Transforms

Published in Wai-Kai Chen, Mathematics for Circuits and Filters, 2000

In most engineering discussions we think of the integrals as Riemann integrals, but in order to handle several convergence questions in the development of Fourier series, convolution theorems, and wavelet transforms, it is necessary to use Lebesgue integration. Lebesgue integration theory has many beautiful results which are not true for the Riemann integral under comparable assumptions about signals. This includes theorems that allow us to interchange limits, integrals, and infinite sums freely.

Rates of return of investments whose timings are specified by a probability distribution

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Journal Information

Published in The Engineering Economist, 2020

Boris Klebanov

The integral in (6) is the Lebesgue integral calculated using the probability measure induced by X (see e.g. Durrett (2010) on the use of Lebesgue integration in probability theory). By using this type of integral one can assume that the random variables Xi are continuous or discrete, or a mixture of both.

Lebesgue integration

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Wavelet Transforms

Rates of return of investments whose timings are specified by a probability distribution