Stability of a conditional Cauchy equation on a set of measure zero

In this paper, we prove the Hyers-Ulam stability theorem when f, g, h : R -> R satisfy vertical bar f(x + y) - g(x) - h(y)vertical bar <= is an element of in a set 1'. R-2 of measure m(1') = 0, which refines a previous result in Chung (Aequat Math 83: 313-320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if f, g, h : R -> R satisfy the Pexider equation f(x +y) - g(x) - h(y) = 0 in Gamma, then the equation holds for all x, y is an element of R. Using our method of construction of the set we can find a set Gamma C R-2n of 2n- dimensional measure 0 and obtain the above result for the functions f, g, h : R-n -> C.