Baire category theorem and approximation of the Pexider quadratic functional equation on a set of Lebesgue measure zero

Let $$\mathbb {R}$$ be the set of real numbers and Y be a Banach space. We investigate the Hyers-Ulam stability theorem when $$f,g:\mathbb {R}\rightarrow Y$$ satisfy the following Pexider quadratic inequality $$\begin{aligned} \Vert f(x+y)+f(x-y)-2g(x)-2f(y)\Vert \le \epsilon , \end{aligned}$$ in a set $$\Omega \subset \mathbb {R}^2$$ of Lebesgue measure $$m(\Omega )=0$$ .