Let $\Omega$$\mathbb B^n$ be a boundedthe open connected subset ofunit ball in $\mathbb R^n$, and $g: \Omega \to \mathbb R^n$$g: \mathbb B^n \to \mathbb R^n$ a measurable function with $|g| \in L^p (\Omega)$$|g| \in L^p (\mathbb B^n)$. Does there exist some function $f$ in the Sobolev space $W^{1, p} (\Omega)$$W^{1, p} (\mathbb B^n)$ such that $\nabla f = g$ a.e.?