I encountered this problem when trying a French agregation subject. I will only mention the relevant parts for my question. I'm not even sure how to tag it, since it contains real analysis, algebra, and it's used to prove a probability result. :)
Denote $L_{1\times 1}$ the space of positive linear forms on the set of measurable functions $h : [0,1]^2 \to \Bbb{R}$ which are bounded bounded (we also know that $\Pi(1)=1$). For each such linear form $\Pi$ we can consider the forms $\Pi_1,\Pi_2$ defined on the set of measurable functions $h:[0,1]\to \Bbb{R}$, which are bounded such that
$ \Pi_1(f)=\Pi(h) \text{ if }h(x,y)=f(x) $
and similar for $\Pi_2$. If there exist densities $\ell_1,\ell_2$ (measurable with integral one) such that $ \Pi_i(f)=\int_0^1 \ell_i(x)f(x)dx$ for every $f$ then we call $\ell_1,\ell_2$ the marginal densities of $\Pi$.
Consider now two densities $q,r$ and $L(q,r)$ the subspace of $L_{1\times 1}$ of forms $\Pi$ with marginal positive densities $q,r$ (recall that the densities are integrable on $[0,1]$ with integral one).
Consider $1_{x\neq y}$ the characteristic function of $\Bbb{R}^2\setminus \{(x,x) : x \in \Bbb{R}\}$.
It is asked to prove that $ \frac{1}{2} \int_0^1 |q-r| \leq \Pi(1_{x\neq y})$
I'm not sure how can I relate $\Pi(1_{x \neq y})$ to the fact that $\Pi$ has marginal densities $q,r$.