ner (1996). We now establish (A.2). Fix a measure Q$_{1}$ over O . Let Q$_{2}$ represent the product measure Q$_{1}$ × P$_{0}$ . Fix ϵ > 0. Let g$_{2}$$_{,}$$_{1}$, . . . , g$_{2}$$_{,m}$ be an ϵ ‖ G$_{2}$ ‖$_{2}$$_{,Q}$$_{2}$ cover of G$_{2}$ under ‖·‖$_{2}$$_{,Q}$$_{2}$ so that min$_{j}$ ‖ g$_{2}$ − g$_{2}$$_{,j}$ ‖$_{2}$$_{,Q}$$_{2}$ < ϵ ‖ G$_{2}$ ‖$_{2}$$_{,Q}$$_{2}$ , where we take m to be equal to its minimal possible value N ( ϵ, Q$_{2}$, G$_{2}$ , G$_{2}$ ). For each j , let g$_{1}$$_{,j}$ ≡ ∫ g$_{2}$$_{,j}$ ( o$_{1}$, o$_{2}$ ) dP$_{0}$ ( o$_{2}$ ). Fix g$_{1}$ ∈ G$_{1}$ . Recall that, by the definition of G$_{1}$ , there exists a g$_{2}$ ∈ G$_{2}$ such that g$_{1}$ ( · ) = ∫ g$_{2}$ ( · , o ) dP$_{0}$ ( o ). Let j ∗ be such that ‖ g$_{2}$ − g$_{2}$$_{,j}$ ∗ ‖$_{2}$$_{,Q}$$_{2}$ ≤ ϵ ‖ G$_{2}$ ‖$_{2}$$_{,Q}$$_{2}$ for this g$_{2}$ . Observe that By the choice of j $^{∗}$, it follows that ‖ g$_{1}$ − g$_{1}$$_{,j}$ ∗ ‖$_{2}$$_{,Q}$$_{1}$ ≤ ϵ ‖ G$_{2}$ ‖$_{2}$$_{,Q}$$_{2}$ = ϵ ‖ G$_{1}$ ‖$_{2}$$_{,Q}$$_{1}$ , where we used that G$_{1}$ ≡ 1 and G$_{2}$ ≡ 1. That is, g$_{1}$$_{,}$$_{1}$, . . . , g$_{1}$$_{,m}$ is an ϵ ‖ G$_{1}$ ‖$_{2}$$_{,Q}$$_{1}$ cover of G$_{1}$ under ‖·‖$_{2}$$_{,Q}$$_{1}$ . Thus, N ( ϵ, Q$_{1}$, G$_{1}$ , G$_{1}$ ) ≤ m . Recalling that we took m = N ( ϵ, Q$_{2}$, G$_{2}$ , G$_{2}$ ), we have shown that N ( ϵ, Q$_{1}$, G$_{1}$ , G$_{1}$ ) ≤ N ( ϵ, Q$_{2}$, G$_{2}$ , G$_{2}$ ). As Q$_{1}$ was arbitrary, for each Q$_{1}$ with support O there exists a Q$_{2}$ with support O 2 such that the the preceding inequality holds. Hence, (A.2) holds, and thus so too does the uniform entropy integral ordering (A.3). For any T ∈ S , we will use the shorthand notation T$_{t}$ ≜ T$_{P}$$_{t}$ , d $_{dt}$T$_{t}$ ∣ t = ˜ t ≜ ˙ T$_{˜}$ t and d 2 dt $_{2}$T$_{t}$ ∣ ∣ t = ˜ t ≜ ¨ T$_{˜}$ $_{t}$. Throughout the appendix we use the following fluctuation submodel through P$_{0}$ for pathwise differentiability proofs: 40 dP$_{t}$ ( o ) ≜ $^{(}$1 + th$_{1}$ ( o ) + t $^{2}$h$_{2}$ ( o ) $^{)}$dP$_{0}$ ( o ) , where P$_{0}$h$_{j}$ = 0 and sup o ∈O | h$_{j}$ ( o ) | < ∞ , j = 1 , 2 . (A.4) Appendix C: proofs ‖ g$_{1}$ − g$_{1}$$_{,j}$ ∗ ‖ 2 2 ,Q$_{1}$ = ∫ ( ∫ [ g$_{2}$ ( o$_{1}$, o$_{2}$ ) − g$_{2}$$_{,j}$ ∗ ( o$_{1}$, o$_{2}$ )] dP$_{0}$ ( o$_{2}$ )$^{)}$$^{2}$ dQ$_{1}$ ( o$_{1}$ ) ≤ ∫ [ g$_{2}$ ( o$_{1}$, o$_{2}$ ) − g$_{2}$$_{,j}$ ∗ ( o$_{1}$, o$_{2}$ )] $^{2}$dQ$_{2}$ ( o$_{1}$, o$_{2}$ ) = ‖ g$_{2}$ − g$_{2}$$_{,j}$ ∗ ‖ 2 2 $_{,Q$_{2}$}$.