Regularity of the graphs of injective additive functions (Q2375918)

If \(X\) is a set with a measure \(\mu\), we say that a set \(A \subset X\) is saturated nonmeasurable with respect to the measure \(\mu\) if neither of the sets \(A\) and \(X \setminus A\) contains a measurable subset of positive measure \(\mu\). If \(X\) is a topological space, we say that a set \(A \subset X\) is saturated non-Baire if neither of the sets \(A\) and \(X \setminus A\) contains a second category subset of \(X\) with the Baire property. Let \(E\) and \(F\) be a non-zero separable (real or complex) Banach spaces and \(\mu: \mathfrak{B}(E) \to [0, +\infty]\), \(\nu: \mathfrak{B}(F) \to [0, +\infty]\) be \(\sigma\)-finite measures vanishing on singletons. Then there exists an injective additive function from \(E\) into \(F\) whose graph is both saturated nonmeasurable with respect to the product measure of \(\mu\) and \(\nu\) and saturated non-Baire subset of \(E\times F\). Let \(E\) be a (real or complex) normed space, card \(E\) = \(\mathfrak{c}\) and \(\mu: \mathfrak{B}(E)\to [0, + \infty]\) be a \(\sigma\)-finite measure. Then there exists a discontinuous and injective additive function from \(E\) into \(\mathbb{R}\) whose graph is both of measure zero (with respect to the completion of product measure of \(\mu\) and Lebesgue measure) and the first category subset of \(E\times F\). The same is true for \(E = \mathbb{R}^{n}\) and \(\mathbb{R}^{m}\) instead of \(\mathbb{R}\).