# How would one extend the Brier score to an infinite number of forecasts?

Is there a neat way to use something like the Brier score to score an infinite set of forecasts/outcomes?

-

A general form of the Brier score, for an essentially arbitrary outcome space $\cal X$, is as follows.
Let $q(\cdot)$ be your quoted density for a random quantity $X$, with respect to a dominating measure $\mu$ over $\cal X$. Then your score, when outcome $X=x$ is realised, is $$S(x, q(\cdot)) = \int q(t)^2 d\mu(t) - 2q(x).$$