In David Schmeidler (1984), "Subjective Probability and Expected Utility without Additivity", "2. Axioms and Background", there is a setting:
Let $X$ be a set and let $Y$ be the set of distributions over $X$ with finite supports
$Y = \{y : X \to [0, 1] \mid y(x) \ne 0 \text{ for finitely many } x\text{'s in } X \text{ and } \Sigma_{x \in X}y(x) = 1 \}$.
Let $S$ be a set and $\Sigma$ be an algebra of subset of $S$. Both sets, $X$ and $S$ are assumed to be nonempty. Denote by $L_0$ the set of all $\Sigma$-measurable finite step functions from $S$ to $Y$ and denote by $L_c$ the constant functions in $L_0$. Let $L$ be a convex subset of $Y^S$ which includes $L_c$. Note that $Y$ can be considered a subset of some linear space and, and $Y^S$, in turn, can be considered as a subspace of the linear space of all functions from $S$ to the first linear space. Whereas it is obvious how to perform convex combinations in $Y$ it should be stressed that convex combinations in $Y^S$ are performed pointwise. I.e., for $f$ and $g$ in $Y^S$ and $\alpha$ in $[0, 1]$, $\alpha f + (1-\alpha g) = h$ where $h(s) = \alpha f(s) + (1-\alpha) g(s)$ on $S$.
I have a hard time understanding/imagining $L_0$, $L_c$, $Y^S$, and $Y$. Can anyone give me a specific examples in understanding their relationships? Thanks!
