Quadratic functional equations in a set of Lebesgue measure zero

Let R be the set of real numbers, Y a Banach space and f:R→Y. We prove the Hyers-Ulam stability theorem for the quadratic functional inequality‖f(x+y)+f(x−y)−2f(x)−2f(y)‖≤ϵ for all (x,y)∈Ω, where Ω⊂R2 is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0.