On Approximately of a \sigma -Quadratic Functional Equation on a Set of Measure Zero

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