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From QED

Let $Q$ be any finite set, and $\mathcal B=2^Q$ be the collection of the subsets of
$Q$. Let $f:\mathcal B\rightarrow \mathbb R$ be a function assigning real numbers to
the subsets of $Q$ and suppose $f$ satisfies the following conditions:
:(i) $f(A)\ge 0$ for all $A\subseteq Q$, $f(\emptyset)=0$,
:(ii) $f$ is monotone, i.e. if $A\subseteq B\subseteq Q$ then $f(A)\le f(B)$,
:(iii) $f$ is submodular, i.e. if $A$ and $B$ are different subsets of $Q$ then
      $$f(A)+f(B)\ge f(A\cap B) + f(A\cup B).\eqno{(2)}$$

Let Q be any finite set, and be the collection of the subsets of $Q$. Let be a function assigning real numbers to the subsets of Q and suppose f satisfies the following conditions:

(i) for all , ,

(ii) $f$ is monotone, i.e. if $A\subseteq B\subseteq Q$ then $f(A)\le f(B)$,

(iii) $f$ is submodular, i.e. if $A$ and $B$ are different subsets of $Q$ then

     Failed to parse (unknown function\leqno): f(A)+f(B)\ge f(A\cap B) + f(A\cup B).\leqno{(2)}

Failed to parse (lexing error): \hat\beta \ \ R_t = \sum^{\infty}_{k=0}\gamma^k r_t{t+k+1} \label{E.eqn0}\\