  In this work we present two particular cases of the general duality result
for linear optimisation problems over signed measures with infinitely many
constraints in the form of integrals of functions with respect to the decision
variables (the measure in question) for which strong duality holds. In the
first case the optimisation problems are over measures with $L^p$ density
functions with $1 < p < \infty$. In the second case we consider a semi-infinite
optimisation problem where finitely many constraints are given in form of
bounds on integrals. The latter case has a particular importance in practice
where the model can be applied in robust risk management and model-free option
pricing.
