The mean has an unbounded directional derivative supremum

By [F1] and [F2], the path law places weight $1-ϵ$ at $0$ and weight $ϵ$ at $a$; this also covers $a=0$, when both atoms coincide.

The identity function is measurable for the Borel sigma-algebra because each Borel set is its own preimage under the identity. Let $S=\{0,a\}$ with its full power-set sigma-algebra and weights $w(b)=F_{ϵ,H}(\{b\})$. Since the complement of $S$ has path-law measure zero, the integral of the identity is the integral of its restriction to this finite probability space: both are the same finite sum over the distinct atoms (apply the simple-function integral separately to the positive and negative parts). Thus [F3] gives $T(F_{ϵ,H})=(1-ϵ)0+ϵa=ϵa$, while $T(F)=0$.

For $ϵ>0$, the difference quotient is $(T(F_{ϵ,H})-T(F))\mathrm{/}ϵ=a$, so the right derivative at zero is $a$. Since $a$ ranges over all real numbers, the supremum of its absolute value is $+\mathrm{\infty }$.