Closed pointwise box of L2 controls

Let $E\subset {\mathbb{R}}^d$ be a bounded Lebesgue-measurable set. Let $U=L^2(E;\mathbb{R})$ with inner product

$$(q,r{)}_U={\int }_{E}q(x)r(x)dx$$

and the corresponding norm. This is a real Hilbert space by $L^2$ with the integral pairing is a Hilbert space under Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

For bounds $q_a,q_b\in U$ with $q_a\le q_b$ almost everywhere, define the admissible box

$$Q_{ad}(q_a,q_b)=\{q\in U:q_a(x)\le q(x)\le q_b(x)\text{ for almost every }x\in E\}\mathrm{.}$$

Equalities and inequalities between controls are understood as almost-everywhere statements about their $L^2$ classes. The box is nonempty because it contains $q_a$. When a finite-element control space is used, its cellwise-constant admissible set is the intersection of this box with the stated piecewise-constant subspace; cellwise averaging preserves the box when both bounds are constant on each control cell.