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At a nonzero optimum, the derivative of expected utility of a proportional transaction cost payoff is zero. Normalized marginal utility therefore prices $X$ at $\epsilon$ for a positive optimum or $-\epsilon$ for a negative optimum. At a zero optimum the one-sided derivative inequalities put $\mathbb{E}X$ inside that interval, so density one works. The competing one-sided-payoff alternative prevents the coercive-maximizer argument from being assumed when a net gain has only one sign.

Under a proportional transaction cost, the sample gain $h_x(\theta )=\theta x-\epsilon |\theta |$ is concave when $\epsilon \ge 0$. The utility function is increasing and concave, soTake finite expectations. Thus $\boxed{G\text{ is concave}}$. The monotonicity of $U$ is essential to composing it with the concave transaction-cost payoff.