On sums of real-valued functions with extremely thick graphs (Q1013798)

Let \({\mathcal E}\) be a family of subsets of a set \(E\). Say that a set \(G \subset E\) is \({\mathcal E}\)-thick if \(G \cap Z \neq \emptyset\) for all \(Z \in {\mathcal E}\). If \(E\) is the product of two separable metric spaces \(E_1\) and \(E_2\), and \({\mathcal E}\) is the collection of Borel sets of the plane \(E_1 \times E_2\) whose projection onto the first coordinate is uncountable, an \({\mathcal E}\)-thick set of the plane is called extremely thick. Generalizing an old result of Sierpiński, the author shows that any function from \({\mathbb R}\) to \({\mathbb R}\) can be written as a sum of two functions whose graphs are extremely thick subsets of the plane \({\mathbb R}^2\). Given a separable metric space \(E\) and a \(\sigma\)-algebra \({\mathcal A}\) of subsets of \(E\) which contains the Borel sets, say that \({\mathcal A}\) is universally extendable if every \(\sigma\)-finite continuous Borel measure on \(E\) can be extended to a measure on \({\mathcal A}\). A measure \(\mu\) on a translation-invariant \(\sigma\)-algebra of subsets of \({\mathbb R}\) is translation-quasi-invariant if the family of \(\mu\)-null sets is preserved under all translations of \({\mathbb R}\). Using the result on sums of functions, the author proves that there are two countably generated, universally extendable, translation-invariant \(\sigma\)-algebras \({\mathcal S}_1\) and \({\mathcal S}_2\) on \({\mathbb R}\) such that there are translation-invariant measures \(\mu_i\) extending \(\lambda\) to \({\mathcal S}_i\) (\(i = 1,2\)) while there is no translation-quasi-invariant extension of \(\lambda\) to the \(\sigma\)-algebra generated by \({\mathcal S}_1 \cup {\mathcal S}_2\). Here \(\lambda\) denotes Lebesgue measure.