[BFJ11, Proposition 8.1]

• (i)

  $P_{\theta}$ is non-decreasing: $f\leq g\Rightarrow P_{\theta}(f)\leq P_{\theta}(g)$.

• (ii)

  $P_{\theta}(f)$ is concave in both arguments:

  $$P_{t\theta+(1-t)\theta^{\prime}}\left(tf+(1-t)g\right)\geq tP_{\theta}(f)+(1-t)P_{\theta^{\prime}}(g)$$

  for $0\leq t\leq 1$.

• (iii)

  For each $c\in\mathbf{R}$ we have $P_{\theta}(f+c)=P_{\theta}(f)+c$.

• (iv)

  $P_{\theta}$ is $1$-Lipschitz continuous, i.e. $\sup_{X}|P_{\theta}(f)-P_{\theta}(g)|\leq\sup_{X}|f-g|$.

• (v)

  Given a bounded function $f$ and a convergent sequence $\theta_{m}\to\theta$ in $N^{1}(\mathcal{X}/S)$ we have $P_{\theta_{m}}(f)\to P_{\theta}(f)$ uniformly on $X$.