The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}} = \mathbb{R} \cup \{-\infty,+\infty\}$.
Convex function is positively homogeneous iff its epigraph is a convex cone. So, to be brief I will call such function a conic function (This is note a standard nomenclature) . If $f$ is conic (and proper, so $f(x) \neq -\infty$) then it is subadditive. Moreover, $f\left(\sum^n_{i=1}\lambda_i v_i\right) \le \sum^n_{i=1} \lambda_i f(v_i)$ for any $v \in V^n,\lambda \in \mathbb{R}^n_{++}$.
This made me wander whether it is possible to extend this to integration in style of Jensen's inequality for arbitrary positive measure $\mu$: $$ (1)\quad f\left(\int_\Omega x(\omega) \; d\mu \right) \le \int_\Omega f(x(\omega)) \; d\mu, $$
where $x : \Omega \to V$ is a function with a vector integral over $\mu$. Inequality (1) actually holds in case $V = \mathbb{R}^n$. See article A Convexity inequality by Paolo Roselli and Michel Willem . Overall, I think I also got an idea how to prove (1) in finite-dimensional case.
So, the natural question is whether this result can be extended to Bochner or Pettis integral for Banach or general topological vector spaces? Clearly (1) holds when $\mu$ has density $\rho$ with respect to some probability measure $\pi$:
$$ \begin{multline} f\left(\int_\Omega x(\omega) \; d\mu \right) = f\left(\int_\Omega x(\omega)\rho(\omega) \; d\pi \right)\le \int_\Omega f\Big(x(\omega)\rho(\omega)\Big) \;d\pi = \\ = \int_\Omega f(x(\omega))\rho(\omega) \; d\pi = \int_\Omega f(x(\omega))d\mu. \end{multline} $$
But in this case $\mu$ must be at most $\sigma$-finite. And in case of $\sigma$-finite $\mu$ there is other ways to prove (1). So, can (1) fail in case of $V$ being an infinite dimensional topological vector space and $\mu$ not being $\sigma$-finite? We may assume that $xd\mu$ is a vector measure of bounded variation, so do we need $V$ to have Radon-Nikodym property? Or in general may there be some additional conditions on $V$? Have you seen any results on this topic?
discussion:
Jochen Wengenroth suggests using linear approximation of conic function $f$. Denote by $\mathcal{L}(f) = \{ g : g\;\text{is linear and}\; g \le f \}$. Then one can write:
$$ \begin{multline} f\left( \int_\Omega x(\omega) \; d\mu\right) = \sup_{g \in \mathcal{L}(f)} g\left( \int_\Omega x(\omega) \; d\mu\right)=_{(2)} \sup_{g \in \mathcal{L}(f)} \int_\Omega g\big(x(\omega)\big) \; d\mu \le_{(3)} \\ \le_{(3)} \int_\Omega \sup_{g \in \mathcal{L}(f)} g\big( x(\omega) \big) \; d\mu = \int_\Omega f\big( x(\omega) \big) \; d\mu \end{multline} $$
Here functions $g$ always exist and (3) is a simple integral majorization which should always work. On the other hand (2) may fail is $g$ is not a closed linear functional. Note, that if we extend such unbounded non-closed linear functional $h$ by $+\infty$, so $f(x) = h(x)$ for $x \in D(h)$ and $f(x) = +\infty$ otherwise. This would be an example of nowhere continuous conic function, where step (2) may fail. However, if $f$ is lower semicontinuous or equivalently has closed convex cone as its epigraph, there always exists a continuous affine minorant. So if I understood everything correctly, I can always get continuous linear functionals to compute (2). So (1) will always hold if $V$ is a Banach space and $f$ is closed.
Roselli, Paolo; Willem, Michel, A convexity inequality., Am. Math. Mon. 109, No. 1, 64-70 (2002). ZBL1037.26019.
