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