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The measurable structure of an exchange economy includes measurability of consumers' preference relations: in particular, for each rational vector $q$, the setis measurable. Continuity of each individual's preference relation alone would not imply this across-consumer requirement. We use this usual measurability convention explicitly.

There are only countably many vectors in ${\mathbb{Q}}^L$. Remove all the null sets $E_q$ that have zero measure, obtainingLet $V=\{q\in {\mathbb{Q}}^L:\mu (E_q)>0\}$. Equivalently, $V$ is the union over $a\in D$ of the rational strict-improvement directions available to consumer $a$. This choice of $D$ ensures that a direction available to even one consumer in $D$ is available to a positive-measure group.

We claim that $0\notin \mathrm{co}V$. Otherwise, take finitely many $q_1,\dots ,q_m\in V$ and positive weights ${\lambda }_i$ with ${\sum }_i{\lambda }_i=1$ and ${\sum }_i{\lambda }_i{q}_i=0$. Choose $t>0$ so small that $2t<{\min }_i\mu (E_{q_i})$. We can choose disjoint measurable sets $F_i\subseteq E_{q_i}$ with $\mu (F_i)=t{\lambda }_i$. Indeed, after the earlier choices remove at most $t$ mass, the remaining portion of $E_{q_i}$ has mass greater than $t$. The scalar case of the Lyapunov convexity theorem, applied to its restricted atomless measure, realizes every mass between zero and its total, including $t{\lambda }_i$.

Now change $f(a)$ to $f(a)+q_i$ on $F_i$ and leave it unchanged elsewhere. Every changed bundle is a feasible consumption bundle and is strictly preferred by its recipient. This new consumption allocation $g$ remains integrable, andThus $g$ is exactly feasible and improves a group of total mass $t>0$, contradicting Pareto efficiency. Notice that no individual's consumption bundles were averaged: an atomless measure allows different small cohorts to make different improvements. Individual convexity of the preference relations is unnecessary.

If $V$ is nonempty, separate its convex hull from zero using the supporting hyperplane theorem. There exists $p\ne 0$ with $p\cdot q\ge 0$ for every $q\in V$. Only weak separation is needed, so it is harmless if zero belongs to the closure of this convex hull.

Fix any $a\in D$ and any $y{\succ }_{a}f(a)$. Completeness and continuity make the strictly preferred set relatively open in ${\mathbb{R}}_{+}^L$. Approximate $y-f(a)$ by rational vectors $q_n$ for which $f(a)+q_n$ is feasible and still strictly preferred. At a zero consumption coordinate, choose the rational approximations from the feasible side. Then $a\in E_{q_n}$ implies $q_n\in V$, and passage to the limit gives $p\cdot (y-f(a))\ge 0$. The same full-measure set $D$ works for all such bundles $y$, not merely for each bundle separately.

If $V$ is empty, the same rational approximation argument says that consumers in $D$ have no strictly preferred feasible bundles at all. In that case any nonzero $p$ satisfies the required implication. In either case,To see why transfers cannot simply be omitted, give all consumers endowment $(1,1)$ and utility function $u(x_1,x_2)=x_1+x_2$. Allocate $(3/2,3/2)$ to half of them and $(1/2,1/2)$ to the other half. This is Pareto efficient, since any Pareto improvement would increase the integral of utility while aggregate resources fix it. At these interior bundles the support inequalities force $p_1=p_2>0$: directions arbitrarily close to $(1,-1)$ force equality, and strict increases force positivity. The poorer half then receives less wealth than the original endowment supplies and has a cheaper strict improvement within its original budget. The support theorem concerns the redistributed budget, as defined in part (b).