Measure zero stability problem of a generalized quadratic functional equation

Let X be a normed space, Y be a Banach space and \(f,g: X\rightarrow Y\). In this paper, we investigate the Hyers–Ulam stability theorem for the generalized quadratic functional equation $$\begin{aligned} f(kx+y)+f(kx-y)=2k^2g(x)+2f(y) \end{aligned}$$ in a set \(\Omega \subset X\times X\), where k is a positive integer. By the Baire category theorem, we derive some consequences of our main result.