Transcript lemma: if the output proposal depends on the member only through a transcript t, then Σ_t Σ_b E_t(b)/r_t(b) ≥ (1/|T|)(Σ_b √Σ_t E_t(b))²

For all $\htmlData{mx=1}{T, J \in \htmlData{mx=0}{\operatorname{Type}}}$, $\htmlData{mx=5}{E : \htmlData{mx=0}{T} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ and $\htmlData{mx=5}{r : \htmlData{mx=0}{T} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=1}{J} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$, if $\htmlData{mx=5}{\forall t, b,\ 0 \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=0}{E}(\htmlData{mx=1}{t}, \htmlData{mx=2}{b})}}$, $\htmlData{mx=5}{\forall t, b,\ 0 \mathrel{\htmlData{mx=4}{<}} \htmlData{mx=3}{\htmlData{mx=0}{r}(\htmlData{mx=1}{t}, \htmlData{mx=2}{b})}}$ and $\htmlData{mx=6}{\forall t,\ \htmlData{mx=4}{\sum_{b} \htmlData{mx=3}{\htmlData{mx=0}{r}(\htmlData{mx=1}{t}, \htmlData{mx=2}{b})}} \mathrel{\htmlData{mx=5}{=}} 1}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{T}}$, $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{T}}$), then