What does "convex class of probability measures" mean in the definition of scoring rules?

Question

Taken from Wikipedia (here), a scoring rule has the following definition

Let $\Omega$ be a sample space, and $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$. Let $\mathcal{P}$ be a convex class of probability measures on $(\Omega, \mathcal{A})$. A scoring rule is a function $S: \mathcal{P} \times \Omega \to \overline{\mathbb{R}}$, where $\overline{\mathbb{R}} := \mathbb{R} \cup \{\pm \infty\}$, such that the integral of $S$ on $\Omega$ exists.

What do they mean by "convex class of probability measures"? Do they mean that the probability measures are convex? If a probability measure are (uniquely) identified by the distribution function, does that measure belong to the class $\mathcal{P}$?

All in all, I am just not sure what "convex class of probability measures" is.

Answer 1

To say that a set $\mathcal{P}$ of probability measures on $(\Omega, \mathcal{A})$ is convex (more standard than saying `a convex class') just means that for all $P, Q \in \mathcal{P}$, and all $\alpha \in [0,1]$, the probability measure $\alpha P + (1-\alpha)Q$ is a member of $\mathcal{P}$.

Recall that $\alpha P + (1-\alpha)Q$ is defined pointwise: for any $A \in \mathcal{A}$, $$(\alpha P + (1-\alpha)Q)(A) := \alpha P(A) + (1-\alpha)Q(A)$$

A bit more generally, convex sets are usually taken to be convex subsets of some ambient vector space, so that convexity is spelled out in terms of the vector space operations.

In this case, the vector space could be the vector space of signed measures on $(\Omega, \mathcal{A})$ with addition and scalar multiplication defined pointwise. So spelling things out, $\mathcal{P}$ is a convex class of probability measures if it contains only probability measures and is a convex subset of that vector space.

Answer 2

What they mean is that for any two probability measure $\mu,\nu$ in $\mathcal P$ and for every $t \in [0,1]$, the probability measure $t\mu + (1-t)\nu$ is also in $\mathcal P$. As $t \in [0,1]$ varies, any probability measure of the form $t\mu + (1-t)\nu$ is known as a convex combination of $\mu$ and $\nu$. So to say that a set of probability measures on $(\Omega,\mathcal A)$ is a complex class of probability measures means that every convex combination of two elements of the set is also an element of the set.

This definition is motivated by the definition of a convex subset of the Cartesian coordinate space $\mathbb R^n$, meaning a subset $P \subset \mathbb R^n$ such that for all points $A,B \in P$ the line segment $\overline{AB}$ is a subset of $P$. The line segment $\overline{AB}$ can be defined, using vector operations on $\mathbb R^n$, as the set of all points of the form $tA+(1-t)B$ for $t \in [0,1]$. In other words a line segment is the set of all convex combinations of its endpoints, and a subset of $\mathbb R^n$ is convex if and only if every convex combination of two points of the subset is also a point of the subset.