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The support statement uses a price quasi-equilibrium with transfers. A pair $(f,p)$ consists of an exactly feasible consumption allocation and a nonzero price vector $p\in {\mathbb{R}}^L$ such that, outside one null set of consumers,The allocation is on the consumer's transfer-adjusted budget boundary, at wealth $m(a)=p\cdot f(a)$. Relative to the original endowment, the lump-sum transfer is $\tau (a)=p\cdot (f(a)-\omega (a))$; exact feasibility gives ${\int }_A\tau d\mu =0$. A price quasi-equilibrium rules out cheaper strict improvements, but may allow a strictly preferred bundle of exactly equal cost. That is why it is weaker than maximizing the preference relation over the entire weak budget set.

Two conventions matter. The conclusion allows redistribution; it does not assert support at each consumer's original wealth $p\cdot \omega (a)$. Also, no monotonicity or free-disposal hypothesis is given, so the nonzero supporting vector is not asserted to have nonnegative coordinates. Those stronger economic conventions require additional hypotheses.

A price quasi-equilibrium with transfers is an exactly feasible consumption allocation $f$ and a nonzero price vector $p$ for which, almost everywhere, every strictly preferred bundle has cost at least $p\cdot f(a)$. The transfer-adjusted wealth is $p\cdot f(a)$; transfers $p\cdot (f(a)-\omega (a))$ sum to zero. Equal-cost strict improvements may remain, which distinguishes a quasi-equilibrium from a utility-maximizing equilibrium. Nonnegative prices need extra assumptions if goods may be undesirable.