Measurability of functionals and of ideal point forecasts

The ideal probabilistic forecast for a random variable $Y$ based on an
information set $\mathcal{F}$ is the conditional distribution of $Y$ given
$\mathcal{F}$. In the context of point forecasts aiming to specify a functional
$T$ such as the mean, a quantile or a risk measure, the ideal point forecast is
the respective functional applied to the conditional distribution. This paper
provides a theoretical justification why this ideal forecast is actually a
forecast, that is, an $\mathcal{F}$-measurable random variable. To that end,
the appropriate notion of measurability of $T$ is clarified and this
measurability is established for a large class of practically relevant
functionals, including elicitable ones. More generally, the measurability of
$T$ implies the measurability of any point forecast which arises by applying
$T$ to a probabilistic forecast. Similar measurability results are established
for proper scoring rules, the main tool to evaluate the predictive accuracy of
probabilistic forecasts.