Riemann integrability and Lebesgue measurability of the composite function
The authors construct a \(2^c-\)dimensional linear space \(V\) of real-valued Riemann integrable functions on \(\mathbb{R}\) and a \(c-\)dimensional linear subspace \(W\) of \(C[0,1]\) such that for every \(f\in V\setminus\{0\}\) and every \(g\in W\setminus\{0\}\) the function \(f\circ g\) is not Riemann integrable. Moreover, they construct a \(c-\)dimensional subspace \(W\) of \(C[0,2]\) with the following property: For every \(g\in W\setminus \{0\}\) there exists a \(c-\)dimensional space \(V\) of measurable functions such that \(f\circ g\) is not measurable for all \(f\in V\setminus \{0\}\).
scientific article; zbMATH DE number 5553669