On a problem of S. Mazur (Q2770604)

In about 1935 S. Mazur asked the following: An additive functional \(f\) in a Banach space \(X\) is given with the property NEWLINE\[NEWLINE\text{for each path \(g\) in \(X,f\circ g\) is Lebesgue measurable}. \tag{P}NEWLINE\]NEWLINE Is it true that \(f\) is continuous? The author proves that, in certain topological spaces, property (P) is equivalent with universal measurability. This makes possible to use ``measurability implies continuity'' type results, that are known for a large class of functional equations, to get ``property (P) implies continuity''.