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Let (N, F) be a down-closed set system. $\forall$ e $\in$ N, there is a non-negative random variable $Z_e$.

I'm confused about what the following notation means:

$I^* \in argmax \{\, \sum_{e \in I} Z_e \mid I \in F \,\}$

I'm a little confused about what kind of entity $I^*$ is. What I think it means is that $I^*$ is a random set which contains each $I \in F$ with some probability. Am I right?

erdoskurdish
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    Can you define what it means to be a "down-closed set system" of random variables? – Gregory Grant Feb 21 '16 at 13:55
  • This means that, for every $\omega$, $I^(\omega)$ is such that $$S(I^(\omega))=\max{S(I,\omega)\mid I\in F}$$ where, for every $I$ in $F$, $$S(I,\omega)=\sum_{e \in I} Z_e(\omega).$$ Thus, if measurable, $I^*$ is a random variable defined on $\Omega$ with values in $F$. – Did Feb 21 '16 at 13:59
  • Let N be your ground set and let F be a family of sets such that $F \subseteq 2^N$. A set system (N, F) is down-closed if, $\forall B \in F$, if $A \subseteq B$, then $A \in F$. – erdoskurdish Feb 21 '16 at 14:02

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