Notation. Finite sets, maps, distributions, random variables, and expectation [ftip-000A]
AGENTDRAFTED
For a finite set \(X\), write \(|X|\) for its cardinality and \(X^*\) for
the set of finite sequences with entries in \(X\). A map \(f:X\to Y\) sends
\(x\in X\) to \(f(x)\in Y\); the inverse image of \(A\subseteq Y\) is
\(f^{-1}(A)=\{x\in X:f(x)\in A\}\).
Write
\[
\Delta (X)=\left \{p:X\to [0,1]:\sum _{x\in X}p(x)=1\right \}
\]
for the probability simplex on \(X\). If \(p\in \Delta (X)\) and
\(Z:X\to \mathbb R\), then
\[
\mathbb E_{x\sim p}[Z(x)]=\sum _{x\in X}p(x)Z(x).
\]
A random variable is a map from the sample set to its value set. Random
variables \(Z\) and \(W\) are independent under \(p\) when
\(p(Z=z,W=w)=p(Z=z)p(W=w)\) for every pair of values \(z,w\).
For nonfinite interaction spaces, probability laws and conditional kernels
require the measurable structure described in Remark [ftip-000C].