Locally Integrable Real Functions

In order to deal with the class of non-integrable real functions we are to discuss here, we must now introduce a different concept of integrability, which is known as local integrability. For integrable real functions in compact domains this is equivalent to both plain integrability and absolute integrability, since all real functions discussed in this paper are assumed to be Lebesgue-measurable functions. Here is the definition, formulated in a way appropriate for our case here.

Definition 1   : A real function $f(\theta)$ is locally integrable on the unit circle if it is integrable on every closed interval contained within that domain.

In this case we say that the real function is locally integrable everywhere on the unit circle. This definition selects the same set of real functions as plain integrability, and also the same set of real functions as absolute integrability, given that all real functions discussed in this paper are assumed to be Lebesgue-measurable functions. This concept of local integrability can now be generalized to functions that have a finite number of non-integrable singularities, giving rise to the concept of local integrability almost everywhere. Given a real function $f(\theta)$ defined on the unit circle, which has a finite number $N\geq 1$ of non-integrable singularities at isolated points on the unit circle, corresponding to the angles $\{\theta_{1},\ldots,\theta_{N}\}$, we introduce the definition that follows.

Definition 2   : A real function $f(\theta)$ is locally integrable almost everywhere on the unit circle if it is integrable on every closed interval contained within that domain, that does not contain any of the points where the function has non-integrable singularities, of which there is a finite number.

This definition begins to characterize the class of non-integrable real functions whose relationship with inner analytic functions we will consider here. Note that we include in this definition the fact that the number of non-integrable singularities must be finite.