Let $\gamma$ be a positive measure on $(\mathbb R^2)^n$ and define a measure $\omega=\sum_{i=1}^n proj^i_\#\gamma$ on $\mathbb R^2$. Notice that
$\omega= \int_{p \in(\mathbb R\times\mathbb R)^n} \alpha_p \,d\gamma(p)$
(or $ \omega= \frac{1}{n} \int_{p \in(\mathbb R\times\mathbb R)^n} \alpha_p \,d\gamma(p)$?)
where $\alpha_p$ is the measure which is uniformly distributed on the set $\{(x_1, y_1), \ldots, (x_n, y_n)\}$ whenever $p=\{(x_i,y_i)\}_{i=1}^n\in(\mathbb R^2)^n$.
My question: how to see that if $\alpha_p^\prime$ is a competitor of $\alpha_p$ for each $p$, and $p\mapsto \alpha_p^\prime$ is measurable, then $\omega^\prime= \int_{p \in(\mathbb R\times\mathbb R)^n} \alpha_p ^\prime\,d\gamma(p)$ is a competitor of $\omega$ ?
Definition(competitor) Let $\alpha$ be a measure on $\mathbb R\times\mathbb R$ with finite first moment in the second variable. We say that $\alpha'$, a measure on the same space, is a competitor of $\alpha$ if $\alpha'$ has the same marginals as $\alpha $ and for $(proj^x_\#\alpha)$-a.e. $x\in\mathbb R$,
$\int y \,d\alpha_x(y)=\int y \,d\alpha_x'(y)$,
where $(\alpha_x)_{x\in\mathbb R}$ and $(\alpha'_x)_{x\in\mathbb R}$ are disintegrations of the measures with respect to $proj^x_\#\alpha$.
Thanks a lot.
That $\omega$ and $\omega^\prime$ have the same marginals is easy to check. It remains to show
$\int_\mathbb R h(x)\int_\mathbb R y \,d\omega_x(y) \,d\omega_1(x)=\int_\mathbb R h(x)\int_\mathbb R y \,d\omega_x^\prime(y) \,d\omega_1^\prime(x)$
for all bounded measurable function $h$, where $\omega_1=\omega_1^\prime$ are the 1st marginal of these measures. This follows from definition-tracing together with the assumption that $\alpha^\prime$ is a competitor of $\alpha$.