Let $n\in\mathbb N$, let $1<p<\infty$, let $U\subset\mathbb R^n$ be bounded and open, and let $u_k,u\in W^{1,p}(U;\mathbb R)$ satisfy $u_k\rightharpoonup u$ in $W^{1,p}(U;\mathbb R)$. Let $(\nu_x)_{x\in U}$ be a family of Borel probability measures on $\mathbb R^n$ generated by the full gradient sequence $(\nabla u_k)_{k=1}^{\infty}$ in the sense that, for every bounded [continuous function](/page/Continuous%20Function) $\psi:\mathbb R^n\to\mathbb R$, the functions $x\mapsto\psi(\nabla u_k(x))$ converge weakly in $L^1(U)$ to $x\mapsto\int_{\mathbb R^n}\psi(\xi)\,d\nu_x(\xi)$. Suppose that $\varphi:\mathbb R^n\to\mathbb R$ is convex and that there exists $C>0$ such that $\varphi(\xi)\le C(1+|\xi|^p)$ for every $\xi\in\mathbb R^n$. Then, for $\mathcal L^n$-a.e. $x\in U$, the integral $\int_{\mathbb R^n}\varphi(\xi)\,d\nu_x(\xi)$ is finite and $\varphi(\nabla u(x))\le\int_{\mathbb R^n}\varphi(\xi)\,d\nu_x(\xi)$.