  In this paper we extend the setting of the online prediction with expert
advice to function-valued forecasts. At each step of the online game several
experts predict a function, and the learner has to efficiently aggregate these
functional forecasts into a single forecast. We adapt basic mixable (and
exponentially concave) loss functions to compare functional predictions and
prove that these adaptations are also mixable (exp-concave). We call this
phenomena integral mixability (exp-concavity). As an application of our main
result, we prove that various loss functions used for probabilistic forecasting
are mixable (exp-concave). The considered losses include Sliced Continuous
Ranking Probability Score, Energy-Based Distance, Optimal Transport Costs &
Sliced Wasserstein-2 distance, Beta-2 & Kullback-Leibler divergences,
Characteristic function and Maximum Mean Discrepancies.
