A characterization of lower semicontinuity (Q2785343)

Let \(\Omega\) be a bounded open set and \(f_0,f_1,\dots,f_k,\dots\) be measurable extended real functions on \(\Omega\) such that \(\{f^-_k\}\subseteq L^1(\Omega)\), then the authors say that the sequence \(\{f_k\}_{k\geq 0}\) satisfies the lower mean value condition at a point \(x_0\in\Omega\) if there exists a null set \(H=H(x_0)\subseteq]0,+\infty[\) such that NEWLINE\[NEWLINE\liminf_{\begin{smallmatrix} h\to0+\\ h\not\in H\end{smallmatrix}} \liminf_{k\to+\infty} {1\over 2^nh^n}\int_{Q_{2h}(x_0)}f_k(x)dx\geq f(x_0),NEWLINE\]NEWLINE where \(Q_{2h}(x_0)\) denotes the cube centred at \(x_0\) and with sidelength \(2h\).NEWLINENEWLINENEWLINEThen they prove that, if there exists \(\lambda\in L^1(\Omega)\) such that \(f_k(x)\geq\lambda(x)\) for a.e. \(x\in\Omega\), the lower mean value condition on \(\Omega\) is equivalent to the inequality NEWLINE\[NEWLINE\liminf_{k\to+\infty}\int_E f_k(x)dx\geq\int_E f_0(x)dxNEWLINE\]NEWLINE for every measurable set \(E\subseteq\Omega\) with \(E^0\neq\emptyset\) and \(\text{meas}(\partial E)=0\).NEWLINENEWLINENEWLINEThe lower mean value condition is also studied and compared to similar notions existing in the literature.