This paper, as advertised, contains none of my own work, but is an attempt to relay some of the flavor of the remarkable solution my friend Miklós Laczkovich has given to a 1925 problem of Alfred Tarski. As most of my knowledge of the history of this problem is directly attributable to Mik, I’ll quote him rather liberally throughout this paper. I’ll begin with the introductory remarks to his paper Equidecomposability and discrepancy; a solution to Tarski’s circle squaring problem which will appear (or perhaps already has appeared) in Crelle’s Journal.

The standard change of variable formula, for $T$ measurable from a measure space $(S,S,\mathrm{\lambda })$ to a measurable space $(\overline{S},\overline{S})$ on which a measurable $\overline{t}$ is defined (see e.g.[H]), ${\displaystyle {\int }_S\overline{t}Td\mathrm{\lambda }}={\displaystyle {\int }_{\overline{S}}\overline{t}d(\mathrm{\lambda }{T}^{-1}})$, is developed for a semigroup valued refinement integral. (The integral on the right can be rewritten in “Jacobian” form, ${\displaystyle {\int }_{\overline{S}}\overline{t}\frac{d\mathrm{\lambda }{T}^{-1}}{d\overline{\mathrm{\lambda }}}}d\overline{\lambda }$, when $\lambda T^{-1}$ is differentiable with respect to a measure $\overline{\mathrm{\lambda }}$ on $\overline{S}$; the attempt in [DS] Lemma III.10.8 to develop the result by starting from $\overline{\mathrm{\lambda }}$ is incorrect). This yields a result for the order-convergent integral of real-valued integrands against positive finitely additive measures in an Archimedian ordered vector space as well as a (correct) result for the [DS] Banach space valued integral of a vector valued integrand against a real-valued finitely additive measure.

We define a variational integral in the m–dimensional Euclidean space so that the Gauss–Green theorem holds for each vector field which is everywhere differentiable (not necessary continuously). The variational integral is then extended by a transfinite sequence of improper integrals, and the Gauss–Green theorem is proved for vector fields which are differentiable only outside fairly large exceptional sets. The variational integral and its extensions are invariant with respect to a continuously differentiable change of coordinates, and hence suitable for integration on differentiable manifolds.

The goal of this paper is to investigate of the semi-Borel and Baire classification of the multifunction of all path derived numbers of a semi-Borel and Baire function of the class $\alpha$. Consequently the classification of the extreme path derivatives is given. The results hold in the setting of ordinary qualitative and approximate path differentiation, and the proofs are based on a classification of the collection of paths which is considered as a multifunction of the semi-Borel class $\alpha$.