The only proof of Jensen inequality (and most general version) that I know is a direct consequence of the Fenchel-Moreau Theorem : If $X$ is a locally convex Hausdorff topological space, let $\mu$ be a Borel probability measure and $x\in X$ be such that for all continuous linear functional $x^\ast\in X^\ast$, $\int_X \langle y,x^\ast\rangle~d\mu=\langle x,x^\ast\rangle$ then we say that $\mu$ averages to $x$, in symbol $\mu\sim x$. The Fenchel-Moreau theorem states that the bidual of a proper l.s.c. convex function $f$ is the function itself. Recall that $f^\ast(x^\ast)=\sup_{x\in X} \langle x,x^\ast \rangle-f(x)$ and $f^{\ast\ast}(x)=\sup_{x^\ast\in X^\ast} \langle x,x^\ast\rangle-f^\ast(x^\ast)$, the theorem states that on $X$, $f=f^{\ast\ast}$.

Suppose that $\mu\sim x$ and $f$ is a proper l.s.c. convex function, then by Fenchel's inequality, for any $y\in X$ and any $x^\ast\in X^\ast$, $\langle y,x^\ast\rangle\leq f(y)+f^\ast(x^\ast)$, taking the integral over $\mu$ we get \begin{align*} \langle x,x^\ast\rangle &= \int_X \langle y,x^\ast\rangle d\mu(y)\\ &\leq \int_X \left[f(y)+f^\ast(x^\ast)\right] d\mu(y)\\ &= \int_X f ~d\mu + f^\ast(x^\ast) \end{align*} and therefore $\langle x,x^\ast\rangle-f^\ast(x^\ast)\leq \int_X f ~d\mu$ for all $x^\ast\in X^\ast$, taking the supremum of the LHS over $x^\ast\in X^\ast$ we get $f(x)=f^{\ast\ast}(x)\leq \int_X f~d\mu$.

I am wondering if the result can be extended to any measurable convex function and if there is any literature on the subject. Is there something like

For any Borel probability measure $\mu$ such that $\mu\sim x\in X$ and any bounded measurable convex functional $f:X\to\mathbb R$, $f(x)\leq \int_X f~d\mu$.

Or is there any reason to believe that this would be false ? Also if true can we generalize to other measurable spaces $X$ where all points can be separated by measurable linear functional ?

I think that I have a hint of proof whenever $X$ is a convex open subset of a Banach space and $f$ is a bounded convex functional, I think this is along the lines of what @MaoWao suggests. I would like to generalize this proof to the case where $X$ is a closed and bounded subset of a Banach space. We prove that in this case $f$ is actually lower semi continuous, by way of contradiction. Suppose that there is $x\in X$ and $x_n\to x$ such that $\liminf_n f(x_n) =f(x)-\delta$ with $\delta>0$ (which we want to contradict). Since $X$ is an open set, there is $\varepsilon>0$ such that $B_{2\varepsilon}(x)\subseteq X$, for any $n$, define $y_n=x-\varepsilon\frac{x_n-x}{\| x_n-x \|}\in B_{2\varepsilon}$, in particular $y_n\in X$. Observe that for any $n$, $x= (1-\alpha_n)x_n + \alpha_n y_n$ with $\alpha_n=\frac{\| x_n-x\|}{\|x_n-x\|+\varepsilon}$ and therefore by convexity, \begin{align*} \Leftrightarrow&&(\| x_n-x\|+\varepsilon)f(x) &\leq \varepsilon f(x_n)+\| x_n-x\| f(y_n)\\ \Leftrightarrow&&\|x_n-x\| f(x) +\varepsilon (f(x)-f(x_n))\leq \|x_n-x\| f(y_n)\\ \Rightarrow && 0<\varepsilon \delta \leq \liminf_n\|x_n-x\| f(y_n)\\ \end{align*} But if $f$ is bounded then the RHS is $0$ which is a contradiction. Now since $f$ is l.s.c. then Jensen inequality applies and we are done.

I am much more interested in the case where $X$ is a closed, convex and bounded subset of Banach space, in this case it feels like a similar argument could be made by working in the largest relatively open subset of $X$ containing $x$, i.e. the largest set containing $x$ in it's relative interior, but there are many point I do not master here, any reference on that would be welcome, the only one I know is Rockafelar for finite dimension.

  • $\begingroup$ It is likely that this question can be challenging. If there's no luck here, you could try on MathOverflow. $\endgroup$
    – Snoop
    Commented Oct 12, 2022 at 11:15
  • $\begingroup$ Do you want any measurable convex function or any bounded measurable function? That makes a huge difference. If a convex function on an arbitrary TVS is only bounded on a non-empty open set, it is continuous everywhere. $\endgroup$
    – MaoWao
    Commented Oct 12, 2022 at 14:47
  • $\begingroup$ @MaoWao In my particular setting I am interested in a bounded measurable convex function on a bounded subset of Banach space. I think that your comment implies for this setting that (even without measurability) the function is lower semi continuous. Is that right ? Is there any reference I could read on the subject ? $\endgroup$
    – P. Quinton
    Commented Oct 12, 2022 at 19:02
  • $\begingroup$ @P.Quinton The subset needs to be open (and convex of course), that's very important for this result. And yes, such a function is necessarily lower semicontinuous, but much more than that, even locally Lipschitz. You can find a proof here: users.mat.unimi.it/users/libor/AnConvessa/continuity_all.pdf. I don't have a good book reference for this kind of questions at hand. $\endgroup$
    – MaoWao
    Commented Oct 13, 2022 at 15:52
  • $\begingroup$ See also here: individual.utoronto.ca/jordanbell/notes/semicontinuous.pdf There are some references, which I cannot check right now. $\endgroup$
    – MaoWao
    Commented Oct 13, 2022 at 15:55


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