What is the most general form of Jensen's inequality? Wikipedia gives for example this more general form, which holds in every topological vector space.

Are there even more general forms, for example dropping continuity assumptions (but keeping measurability/integrability)?

Or are there maybe counterexamples, by which I mean Jensen-like statements which fail if the conditions are too lax?

Any reference would also be welcome.

(Are the tags appropriate?)

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    $\begingroup$ One note is that is the only properties of the integral that get used are that it's linear and positivity preserving. So you also sometimes see Jensen used for other such operators that don't necessarily have anything to do with integration - it's a theorem of algebra, not measure theory. I am not sure what the most general formulation of this is - probably something to do with ordered vector spaces. $\endgroup$ – Nate Eldredge Nov 15 '19 at 17:45
  • $\begingroup$ In some sense, rather than trying to formulate a Grand Unified Jensen Inequality, it's probably more useful day-to-day to just remember the proof and what ingredients it does and doesn't use. $\endgroup$ – Nate Eldredge Nov 15 '19 at 17:47
  • $\begingroup$ @NateEldredge What would be an example with other operators? $\endgroup$ – geodude Nov 15 '19 at 18:24
  • $\begingroup$ @geodude: Oh, I left out one property that's used: $\int 1 =1$. So it's used for so-called Markovian operators. Let $X$ be some vector space of real-valued functions that contains the constant $1$, let $P : X \to X$ be a linear positivity-preserving operator with $P1=1$ (i.e. Markovian), and let $\varphi$ be convex. Then if $f, \varphi \circ f \in X$, this version of Jensen says that $P[\varphi \circ f] \ge \varphi(Pf)$ pointwise. A typical example is where $P = P_t$ is the heat semigroup acting on $L^\infty$. $\endgroup$ – Nate Eldredge Nov 16 '19 at 9:10

A generalized Jensen inequality for a conditional expectation of a random function (with values in an ordered Banach space) of a random vector in another Banach space is given here, with some review of the prehistory.

The proof does not seem trivial; it takes over three pages and contains references to previous results.

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