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An exchange economy specifies consumers, their preference relations, their consumption sets and their initial endowments. A consumption allocation is feasible when aggregate consumption equals aggregate endowment; free-disposal models instead allow a coordinatewise inequality. These conventions must be distinguished when specifying supporting prices.

The range of a finite-dimensional, finite, countably additive, atomless vector measure is compact and convex. In particular, an atomless measure restricted to a measurable set $E$ can realize every mass between zero and $\mu (E)$. This allows exact balancing by small disjoint consumer cohorts in an exchange economy.

Use the exact-resource-balance convention for an exchange economy. If $\omega$ is the integrable initial-endowment profile, a feasible consumption allocation is a measurable integrable function $f:A\to {\mathbb{R}}_{+}^L$ withIt is Pareto efficient if there is no other feasible consumption allocation $g$ satisfying $g(a){⪰}_{a}f(a)$ for almost every consumer and $g(a){\succ }_{a}f(a)$ on a set of positive measure. A Pareto improvement makes nobody worse off, up to a null set, and makes a positive-measure group strictly better off. Positive measure is essential: reallocating to a single consumer of zero mass does not constitute an aggregate Pareto improvement.

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).

An exchange economy is regular when the derivative of its reduced aggregate excess demand has full rank at every equilibrium. For two goods and normalized prices $(p,1)$, this means $Z_p(p)\ne 0$ at every zero of first-good excess demand. Regular equilibria are locally isolated and persist smoothly under small parameter changes.