 Logarithmic terminal wealth in a complete market

In a complete market with a unit cash asset and normalized strictly positive state-price density $Z$, the optimal positive terminal portfolio wealth is $x/Z$ whenever it is replicable in the admissible class. The state-price budget constraint is an equality: $\mathbb{E}[Z(x/Z)]=x$. The utility conjugate inequality gives $\mathbb{E}\log H\le \log x-\mathbb{E}\log Z$ for every affordable positive claim. This value can be infinite, because $\mathbb{E}(\log Z{)}^{+}<\infty$ while $\mathbb{E}(\log Z{)}^{-}$ need not be finite. Even completeness for bounded claims suffices to approximate the value: set $b_n=\mathbb{E}[Z/(Z\vee n^{-1})]$ and $H_n=x/[b_n(Z\vee n^{-1})]$. Each claim is bounded and costs $x$, while its expected logarithmic utility tends to the displayed value by monotone convergence after an integrable lower bound is subtracted.