Definition 21.11.1561B746F5E884

Definition 21.11.15.61B746F5E884 Let $F\in \mathrm{Conv}(N)$. The closure $\operatorname{cl}F\in \mathrm{Conv}(N)$ of $F$ is defined as follows: If $F(n)=-\infty $ for some $n\in N$, then $\operatorname{cl}F\coloneqq -\infty $. Otherwise, we define $\operatorname{cl}F$ as the lower semicontinuity regularization of $F$.

A convex function $F\in \mathrm{Conv}(N)$ is closed if $F=\operatorname{cl}F$. In other words, $F\in \mathrm{Conv}(N)$ is closed if one of the following conditions holds:

1. 9B2264291DE3

   $F\equiv -\infty $;

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   $F\equiv \infty $;

3. C1225E82131E

   $F$ is proper and lower semicontinuous.