Assume that $J=J\big(x_1,x_2\big)$ is a locally bounded function from $\mathbb R^2\to \mathbb R$ and $J_n=J_n\big(x_1,x_2\big)$ is a sequence of functions from $\mathbb R^2\to \mathbb R$ that locally uniformly converges to $J$.
Fix $B$ a bounded subsets of $\mathbb R^2$ and $\ell\in\mathbb N$. I want to show that the following convergence holds \begin{equation}\label{eq:conv_goal} \int_{B^\ell} \det\left(\left[J_n\big(x_i,x_j\big)\right]_{1\leq i,j \leq \ell}\right) dx_1\dots dx_\ell \to \int_{B^\ell} \det\left(\left[J\big(x_i,x_j\big)\right]_{1\leq i,j \leq \ell}\right) dx_1\dots dx_\ell. \end{equation} We start by fixing $\bar x\in B^\ell$. Since by assumption $J_n$ locally uniformly converges to $J$ and $J$ is locally bounded, there exists a neighborhood $U_{\bar x}$ of $\bar x$ such that for all $n\in\mathbb N$, \begin{equation}\label{eq:dom} \det\left(\left[J_n\big(x_i,x_j\big)\right]_{1\leq i,j \leq \ell}\right)\leq \prod_{j=1}^\ell \Big(\sum_{i=1}^\ell J_n\big(x_i,x_j)^2\Big)^{1/2}\leq C, \quad\forall (x_i)_{1\leq i\leq \ell}\in U_{\bar x}, \end{equation} where in the first inequality we used Hadamard's inequality. Now note that since $J_n$ locally uniformly converges to $J$, then for every fixed $(x_i)_{1\leq i\leq \ell}\in B^\ell$, \begin{equation*} \det\left(\left[J_n\big(x_i,x_j\big)\right]_{1\leq i,j \leq \ell}\right) \to \det\left(\left[J\big(x_i,x_j\big)\right]_{1\leq i,j \leq \ell}\right) \end{equation*} Now I want to use dominated convergence to conclude but my dominations holds only locally. Any help to conclude the proof?