Definiton of inner product on $L^2$ with positive definite symmetric matrix

Question

Consider a measure space $\left(\Omega,\mathcal A,\mu\right)$. Now one can define an inner product $$\langle f,g\rangle_Q:=\int_\Omega\int_\Omega f(x)g(y)q(x,y)\mu\left(dx\right)\mu\left(dy\right),$$ where $q$ is symmetric and positve definite in the sene that $\forall A_1,\ldots A_n\in\mathcal A$ the matrix $Q$ with elements $q_{i,j}=\int_{A_i}\int_{A_j}q(x,y)\mu(dx)\mu(dy)\ge0$ is symmetric and positve definite.

Does this definition have to be understood in the following way. We take $q_{ij}$ as given and now construct $q(x,y)$ such that above equation for every measurable set holds true? I am not sure if $q$ is now well defined (i.e unique).

I really appreciate some enlightening on this defintion.

Answer 1

The $q$ is given. The matrix with entries $q_{ij}$ is used to establish a property that $q$ satisfies.