As Steve Huntsman points out in his comment this is just writing out the integral against the Hausdorff measure on the fractal domain in polar coordinates over the same fractal. Since the author does not specify what the fractal domain actually and merely asserts that it is distributed throughout $\mathbb{R}^{4}$ such that it is "regular and homogenous" with respect to the Hausdorff measure he is able to claim that for the right sort of fractal domain the formulas that we are used to when integrating over $\mathbb{R}^{d}$ hold even when $d$ is not an integer. That is what looks like the tricky part to me, just assuming the existence of such a fractal. The discussion of why the coefficient is what it is appears just after (2.6) and arises simply by analogy to the volume of the unit sphere that usually appears in polar coordinate transforms.