Continuity of some non-local functionals with respect to a convergence of the underlying measures

We study some non-local functionals on the Sobolev space $W^{1,p}_0(Ω)$ involving a double integral on $Ω\timesΩ$ with respect to a measure $μ$. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of $Γ$-convergence of the corresponding functionals.

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