Continuous-space extension of the entropy duality lemma

Establish a version for continuous type spaces Z of the equality proved in Lemma LemmEquivLemm5.2Chatterjee, namely that the relative entropy of a probability graphon with respect to a reference probability measure equals the supremum of the functionals J_{A,\nu} over the specified class of admissible kernels A.

Background

The paper proves in Lemma LemmEquivLemm5.2Chatterjee that, when the type space Z is finite, the probability-graphon relative entropy admits a variational representation as the supremum of the functionals J_{A,\nu}. This identity is used to establish lower semicontinuity of the entropy and to support the large-deviation upper bound.

The authors explain that their treatment first handles finite type spaces and then extends the large deviation principle to compact Polish spaces through the Dawson–Gärtner theorem. They explicitly leave unresolved whether the finite-dimensional variational identity itself has an analogue for continuous Z, noting that the relevant dual-space identification becomes difficult in the compact-space setting.