= Solution
Interpret the invertibility assumption on $G$ as continuity and strict increase on the relevant range, so that $\nu$ is an <atomless measure>. Suppose a non-decreasing <transport map> $T$ with $T_\#\mu=\nu$ exists. Then $\mu$ is also an <atomless measure>: if $\mu(\{x\})>0$, the <pushforward measure> would give $\nu(\{T(x)\})\geq\mu(\{x\})>0$. Thus $F$ is continuous.
At any point $x$ where the non-decreasing representative is defined, the definition of a <monotone function> gives
$$
\{z:z\leq x\}\subseteq\{z:T(z)\leq T(x)\},\qquad
\{z:T(z)<T(x)\}\subseteq\{z:z<x\}.
$$
Using the <pushforward measure> identity and continuity of the two <cumulative distribution functions>, we obtain
$$
F(x)\leq G(T(x)),\qquad
G(T(x))=G(T(x)-)\leq F(x-)=F(x).
$$
Consequently
$$
\boxed{T(x)=G^{-1}(F(x))\quad\mu\text{-almost everywhere}.}
$$
The <monotone rearrangement> in part (c) is optimal for the <convex> difference cost, so $T$ has the same cost and solves the <Monge optimal transport problem>. The meaningful uniqueness is up to a $\mu$-null set; arbitrary values away from the source do not affect transport or cost. Existence of the non-decreasing <transport map> supplies the source condition missing in part (c).
Back to article page
