= Solution
For a cost $c:X\times Y\to[0,+\infty]$ that is a <Borel measurable function>, a <transport map> is a measurable $T:X\to Y$ whose <pushforward measure> satisfies
$$
T_\#\mu=\nu,\qquad \mu(T^{-1}(B))=\nu(B)\quad\text{for every Borel }B\subseteq Y.
$$
The <Monge optimal transport problem> moves every source point to one destination:
$$
\boxed{\inf_{T_\#\mu=\nu}\mathbb M(T),\qquad \mathbb M(T)=\int_X c(x,T(x))\,d\mu(x).}
$$
The <Kantorovich optimal transport problem> permits mass to split. Its admissible <transport plans> are the <probability measures> on $X\times Y$ with prescribed <marginal distributions>:
$$
\Pi(\mu,\nu)=\{\pi:(p_X)_\#\pi=\mu,\ (p_Y)_\#\pi=\nu\},\qquad
\boxed{\inf_{\pi\in\Pi(\mu,\nu)}\mathbb K(\pi),\quad \mathbb K(\pi)=\int_{X\times Y}c\,d\pi.}
$$
Thus a <transport plan> is a <coupling of probability distributions>. The set $\Pi(\mu,\nu)$ is never empty: it contains the <product measure> $\mu\otimes\nu$. Signed costs can also be used when their integrals are well defined, for example with an integrable lower bound of the form $a(x)+b(y)$.
On the <Polish space> $\mathbb R$, take the <Dirac measures>
$$
\boxed{\mu=\delta_0,\qquad \nu=\tfrac12\delta_{-1}+\tfrac12\delta_1.}
$$
Every measurable map satisfies $T_\#\delta_0=\delta_{T(0)}$, which cannot equal $\nu$. A <transport map> cannot split an <atom of a measure>, whereas the <transport plan> $\tfrac12\delta_{(0,-1)}+\tfrac12\delta_{(0,1)}$ can.
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