On every affine line in $C(X)$, the concave function $B_\varphi $ is differentiable outside a countable set, so the gap between its two directional derivatives along the line vanishes on a dense subset. Upper semicontinuity of this gap and separability of $C(X)$ turn the one-dimensional fact into uniqueness of the supporting probability on a dense $G_\delta $ subset of $C(X)$.
Theorem 11.34.7.A4850266BE73 The set
\[ \mathscr U_\varphi =\{ F\in C(X,\mathbb R):\mathcal A_\varphi (F) \text{ is a singleton}\} \]
is a dense $G_\delta $ subset of $C(X)$ with the uniform norm. At every $F\in \mathscr U_\varphi $, the functional $B_\varphi $ is Gâteaux differentiable, and its derivative is minus the integral against the unique supporting probability. If $F_ j\to F$ uniformly and $\nu _ j\in \mathcal A_\varphi (F_ j)$, then $\nu _ j$ converges weakly to that probability.
Proof.
For $h\in C(X,\mathbb R)$ define the width
\[ w_ h(F)=\max _{\nu \in \mathcal A_\varphi (F)}\int _ Xh\, \mathrm{d}\nu -\min _{\nu \in \mathcal A_\varphi (F)}\int _ Xh\, \mathrm{d}\nu \geq 0. \]
The extrema exist by weak compactness. We claim that the width is upper semicontinuous in $F$. Indeed, let $F_ j\to F$ and choose maximizing and minimizing probabilities at each $F_ j$. Along a subsequence realizing the upper limit of the widths, compactness of the space of probability measures gives weak limits of both choices. By proposition 11.34.3, both limits belong to $\mathcal A_\varphi (F)$, so the difference of their integrals against $h$ is at most $w_ h(F)$.
Consequently, for every $\epsilon >0$ the set $\{ F:w_ h(F)<\epsilon \} $ is open. It is also dense. If $h=0$ there is nothing to prove. Otherwise fix $F$ and an open ball about it. The function
\[ b(t)=B_\varphi (F+th) \]
is finite and concave on the real line, so it is differentiable outside an at most countable set: its one-sided derivatives are finite and monotone, and the nonempty open intervals between unequal one-sided derivatives at distinct points are disjoint, so each contains a different rational number. Choose a point of differentiability $t$ so close to zero that $F+th$ lies in the given ball. The derivative formulas of proposition 11.34.5 give $w_ h(F+th)=0$, which proves density.
The compact manifold $X$ is metrizable, so $C(X)$ is separable. Choose a countable uniformly dense family $(h_ j)$ in $C(X)$. By the Baire category theorem in the complete metric space $C(X)$, the set
\[ \bigcap _{j,k\geq 1}\{ F:w_{h_ j}(F)<1/k\} \]
is a dense $G_\delta $. At every point of this intersection, all supporting probabilities have the same integral against every $h_ j$. Since they have unit mass, uniform approximation gives equal integrals against every continuous function, and the Riesz representation theorem shows that the probabilities coincide. Conversely, a singleton has zero width in every direction, so the intersection is exactly $\mathscr U_\varphi $. The formula for the derivative follows from proposition 11.34.5, and the approximation assertion from proposition 11.34.3.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.