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**nylinka** Dear mathematicians,

I would like to ask you if anybody of you could explain me the proof of the theorem of semicontinuity:

Let $\displaystyle \Omega$ be an open set in $\displaystyle R^n$, let M be a closed set in $\displaystyle R^N$, and let $\displaystyle F(x, u, z)$ be a function defined in $\displaystyle \Omega \times M \times R^\upsilon$ and such that

(i) $\displaystyle F$ is a CARATHEODORY function, that is measurable in x for every $\displaystyle (u,z)\in M \times R^\upsilon$ and continuous in $\displaystyle (u,z)$ for almost every $\displaystyle x\in \Omega$,

(ii) $\displaystyle F(x, u, z)$ is convex in $\displaystyle z$ for almost every $\displaystyle x \in \Omega$ and for every $\displaystyle u\in S$,

(iii) $\displaystyle F\geq 0$.

Let $\displaystyle u_h, u \in L^1(\Omega,M),\ z_h,z \in L^1(\Omega, R^\upsilon)$ and assume that $\displaystyle u_h\to u$ and $\displaystyle z_h \to z$ in $\displaystyle L_{loc}^1 (\Omega)$. Then,

$\displaystyle \displaystyle\int_{\Omega} F(x, u, z)dx\le \liminf_{h \to \infty}\int_{\Omega}F(x,u_h,z_h)dx$.