= Solution
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,
$$
\boxed{y\in\mathbb R_+^L,\quad y\succ_a f(a)\quad\Longrightarrow\quad p\cdot y\geq p\cdot f(a).}
$$
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.
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