On the existence of certain singular integrals

Let f (x) and K (x) be two functions integrable over the interval (-∞, +∞). It is very well known that their composition - ^{ + } f (t) K ({x - t) dt} exists, as an absolutely convergent integral, for almost every x. The integral can, however, exist almost everywhere even if K is not absolutely integrable. The mostinteresting special case is that of K (x) = 1/x. Let us set f (x) = 1 - ^{ + } {f (t) {x - t}dt}.