 Solution

By part (a), $f$ is a bounded continuous function, hence is Borel measurable and integrable for the given probability measure. By part (e), $l\in E^{\prime }$, and its restriction to the unit ball is also bounded and integrable. For every $u\in U$, the supporting inequality says $f(u)\ge f(x)+l(u)-l(x)$. Taking expectations and using the barycentre hypothesis for this particular $l$ givesThus $\boxed{\mathbb{E}f\ge f(x)}$, the Jensen inequality. This proof only uses the stated weak barycentre identity; a vector-valued integral for the identity map is not needed.