Chapter 8, Theorem 1 — finite-partition Jensen lower bound

Fix the following data throughout.

• $\Omega$ is a type carrying a $\sigma$-algebra, and $\mu$ is a measure on $\Omega$ which is assumed to be a probability measure ($\mu(\Omega)=1$).
• $E$ is a real Banach space: a normed abelian group with a real normed-space structure, assumed complete.
• $\alpha$ is an arbitrary type with no structure at all.
• $\Xi \subseteq E$ is a subset assumed to be convex (over $\mathbb R$) and closed.
• $\xi : \Omega \to E$ is a function such that
  • for $\mu$-almost every $\omega$, $\xi(\omega) \in \Xi$ (no requirement that $\xi(\omega)\in\Xi$ for every $\omega$, and $\xi$ is not separately assumed measurable except through the next item), and
  • $\xi$ is Bochner integrable with respect to $\mu$ (which includes being a.e.-strongly measurable and having finite integral of $\|\xi\|$).
• $D \subseteq \alpha$ is a set and $x \in \alpha$ is a point with $x \in D$. Neither $D$ nor the membership $x\in D$ is used anywhere else in the statement; the hypothesis $x \in D$ imposes nothing on $g$, $\xi$, or the conclusion.
• $g : \alpha \to E \to \mathbb R$ is a function of two arguments; write $g_x := g(x,\cdot) : E \to \mathbb R$. It is assumed that
  • $g_x$ is convex on $\Xi$ (the set $\Xi$ is convex and, for $y,z\in\Xi$ and $a,b\ge 0$ with $a+b=1$, $g_x(ay+bz)\le a\,g_x(y)+b\,g_x(z)$); nothing is assumed about $g_x$ outside $\Xi$;
  • $g_x$ is continuous on $\Xi$ (continuous when restricted to $\Xi$ with the subspace topology);
  • the real-valued function $\omega \mapsto g_x(\xi(\omega))$ is integrable with respect to $\mu$.
• $\nu$ is a natural number, possibly $0$, and $P$ is an element of the type $\mathrm{Partition}\,\mu\,\nu$.

The type $\mathrm{Partition}\,\mu\,\nu$ and its two projections used below are not defined in the code shown; they come from the imported file Definitions.Def_StochasticProg_Bounds_Partition, whose contents are not available here. All that can be read off from this declaration is their usage:

• $P.\mathrm{weight}$ is a family of real numbers $p_l$ indexed by some finite index type over which $\sum_l$ ranges (presumably $l \in \{0,\dots,\nu-1\}$); the statement itself does not assert that $p_l \ge 0$ or that $\sum_l p_l = 1$ — whatever constraints hold are those built into the definition of $\mathrm{Partition}$, which is not visible.
• $P.\mathrm{condMean}\,\xi\,l$ is an element $\xi^{(l)}$ of $E$, produced from $\xi$ and the block index $l$ by a construction internal to $\mathrm{Partition}$. Whether this is a genuine conditional expectation of $\xi$ on the $l$-th block, what value it takes if a block has measure zero (a total function must return something, possibly a junk value such as $0$), and whether it lies in $\Xi$, are all determined by the unseen definition and not asserted by this theorem.

Under all of the above hypotheses, the theorem asserts the single real inequality

where the left side is a finite sum over the index set of $P$, with $p_l = P.\mathrm{weight}\,l$ and $\xi^{(l)} = P.\mathrm{condMean}\,\xi\,l$, and the right side is the Bochner (here: Lebesgue) integral of the real function $\omega\mapsto g_x(\xi(\omega))$ against $\mu$.

Edge cases silently included:

• If $\nu = 0$ and the index set is therefore empty, the left side is the empty sum $0$, and the claim reduces to $0 \le \int g_x(\xi)\,d\mu$.
• The inequality is non-strict ($\le$), and it is a single inequality, not an equality or a two-sided bound.
• Since the hypotheses on $\xi$ are only almost-everywhere membership in $\Xi$, the values $g_x(\xi(\omega))$ on the $\mu$-null exceptional set are unconstrained by convexity or continuity but do not affect the integral.
• If $\xi^{(l)} \notin \Xi$ for some $l$, then $g_x(\xi^{(l)})$ is a value of $g_x$ at a point where no convexity or continuity assumption applies; the statement still evaluates $g_x$ there.
• If $\Xi$ is empty, the hypothesis "$\xi(\omega)\in\Xi$ for a.e. $\omega$" is unsatisfiable for a probability measure, so the theorem is vacuous in that case.