Abstract:
In the same paper Titchmarsh established that, when studying the properties of trigonometric series conjugate to Fourier series of Lebesgue integrable functions, $Q$-integration leads to a series of natural results. A very uncomfortable fact impeding the application of $Q$-integrals and $Q'$-integrals when studying diverse problems of function theory is the absence of the additivity property. If one adds the some condition to the definition of $Q$-integrability ($Q'$-integrability) of a function $f$, then the $Q$-integral and $Q'$-integral become additive. In this paper, we give the definition of $Q$- and $Q'$-integrals for the function, measurable on the real axis $R$, and study its additivity properties.
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