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 ) − 2 f ( x ) − 2 f ( y ) ‖ ≤ ϵ for all ( x , y ) ∈ Ω , where Ω ⊂ R 2 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.