Foundational aspects of singular integrals

Abstract: We investigate integration of classes of real-valued continuous functions on (0,1]. Of course difficulties arise if there is a non-$L^1$ element in the class, and the Hadamard finite part integral ({\em p.f.}) does not apply. Such singular integrals arise naturally in many contexts including PDEs and singular ODEs. The Lebesgue integral as well as $p.f.$, starting at zero, obey two fundamental conditions: (i) they act as antiderivatives and, (ii) if $f =g$ on $(0,a)$, then their integrals from $0$ to $x$ coincide for any $x\in (0,a)$. We find that integrals from zero with the essential properties of $p.f.$, plus positivity, exist by virtue of the Axiom of Choice (AC) on all functions on $(0,1]$ which are $L^1((\epsilon,1])$ for all $\epsilon>0$. However, this existence proof does not provide a satisfactory construction. Without some regularity at $0$, the existence of general antiderivatives which satisfy only (i) and (ii) above on classes with a non-$L^1$ element is independent of ZF (the usual ZFC axioms for mathematics without AC), and even of ZFDC (ZF with the Axiom of Dependent Choice). Moreover we show that there is no mathematical description that can be proved (within ZFC or even extensions of ZFC with large cardinal hypotheses) to uniquely define such an antiderivative operator. Such results are precisely formulated for a variety of sets of functions, and proved using methods from mathematical logic, descriptive set theory and analysis. We also analyze $p.f.$ on analytic functions in the punctured unit disk, and make the connection to singular initial value problems.