Let $(\Omega,\mathcal F,\mathbb P)$ be a [probability space](/page/Probability%20Space). Let $X:(\Omega,\mathcal F)\to(\{0,1\},2^{\{0,1\}})$ be a binary treatment [random variable](/page/Random%20Variable), and let $Y,Y_0,Y_1:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be the observed outcome and the two potential-outcome random variables. Assume consistency: for each $a\in\{0,1\}$, $Y=Y_a$ on $\{X=a\}$. Assume marginal exchangeability: for each $a\in\{0,1\}$, the random variable $Y_a$ is independent of $X$. Fix $x\in\{0,1\}$ with $\mathbb P(X=x)>0$. Then, for every Borel set $A\in\mathcal B(\mathbb R)$, $\mathbb P(Y_x\in A)=\mathbb P(Y\in A\mid X=x)$. Consequently, if $Y_x\in L^1(\Omega,\mathcal F,\mathbb P)$, then $Y$ is integrable conditionally on $\{X=x\}$ and $\mathbb E[Y_x]=\mathbb E[Y\mid X=x]$.