**Appendix C**

**Proof of Corollary 2.** The set <sup>W</sup>(F*L*|*y*) ∈ B(R+1) is compact, and <sup>W</sup>(F*L*|*y*) <sup>⊆</sup> conv(W(C|*y*)). By the Krein–Milman theorem [37], *each* point of the set <sup>W</sup>(F*L*|*y*) can be represented as a convex combination at most of dim(W(F*L*|*y*)) <sup>+</sup> 1 extreme points of the set <sup>W</sup>(F*L*|*y*).

Obviously, all extreme points of <sup>W</sup>(F*L*|*y*) belong to the set <sup>W</sup>(C|*y*). Hence, for the point **<sup>w</sup>** (*y*) which is a solution to the finite-dimensional dual problem (28), there exists a finite set {**q***s*(*y*)}*s*=1,*<sup>S</sup>* ⊆ C, 1 *<sup>S</sup>* dim(W(F*L*|*y*)) <sup>+</sup> 1 of parameters, and weights {P*s*(*y*)}*s*=1,*<sup>S</sup>* (P*s*(*y*) - 0, ∑*<sup>S</sup> <sup>s</sup>*=<sup>1</sup> P*s*(*y*) = 1) such that:

$$
\widehat{\mathbf{w}}(y) = \sum\_{s=1}^{S} \mathbf{P}\_s(y) w(\mathbf{q}\_s(y)|y). \tag{A47}
$$

The parameters and weights define the reference measure (15) on the space (C, B(C)):

$$
\hat{F}'(dq|y) \stackrel{\triangle}{=} \sum\_{s=1}^{S} \mathbb{P}\_s(y)\delta\_{\mathbf{q}\_s(y)}(dq).
$$

We can establish the initial measure by (16):

$$
\hat{F}(dq|y) = \frac{\sum\_{s=1}^{S} \mathcal{L}^{-1}(\mathbf{q}\_s(y)|y) \mathsf{P}\_s(y) \delta\_{\mathbf{q}\_s(y)}(dq)}{\sum\_{s'=1}^{S} \mathcal{L}^{-1}(\mathbf{q}\_{s'}(y)|y) \mathsf{P}\_{s'}(y)}.
$$

It is easy to verify that <sup>E</sup>*F* ? *h*(*γ*, *X*)-<sup>2</sup>|*<sup>Y</sup>* <sup>=</sup> *<sup>y</sup>* @ <sup>=</sup> **<sup>w</sup>**1(*y*) and <sup>E</sup>*F*{*h*(*γ*, *<sup>X</sup>*)|*<sup>Y</sup>* <sup>=</sup> *<sup>y</sup>*} <sup>=</sup> **<sup>w</sup>**2(*y*), i.e., *<sup>F</sup>* is the required LFD. Corollary <sup>2</sup> is proven.
