Let
$X$
be a real normed space,
$Y$
a Banach space, and
$f\,:\,X\,\to \,Y$
. We prove theUlam–Hyers stability theorem for the cubic functional equation
$$f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right)\,=\,0$$
in restricted domains. As an application we consider a measure zero stability problem of the inequality
$$\left\| f\left( 2x\,+\,y \right)\,+\,f\left( 2x\,-\,y \right)\,-\,2f\left( x\,+\,y \right)\,-\,2f\left( x\,-\,y \right)\,-\,12f\left( x \right) \right\|\,\le \,\varepsilon$$
for all
$\left( x,\,y \right)$
in
$\Gamma \,\subset \,{{\mathbb{R}}^{2}}$
of Lebesgue measure 0.