Locally integrable functions as regular distributions

Assume Countable Choice for the published injection theorem. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, with $n\ge 1$. For $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$, its regular distribution is $T_u(\phi ):={\int }_{\mathrm{\Omega }}u(x)\phi (x)dx,\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })\mathrm{.}$ The integral is finite because $\phi$ is bounded and has compact support. The pairing is complex bilinear: there is no conjugation. It depends only on the almost-everywhere class of $u$, and the induced map $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })\mathrm{/}\text{\hspace{0.17em}⁣}{\sim }_{\mathrm{a}\mathrm{.}\mathrm{e}\mathrm{.}}⟶{\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega }),[u]⟼T_u$ is complex-linear and injective. The pairing is the regular functional of Regular distribution from a locally integrable function, and the injection is Locally integrable functions embed in distributions under The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Countable Choice is used only for that published injectivity theorem; defining the pairing and its linearity require no choice. If $\mathrm{\Omega }=\mathrm{\varnothing }$, both spaces contain only zero and the map is the unique injection.

Well-definedness. If $u=\tilde{u}$ almost everywhere, then for each test function the integrands agree almost everywhere, so their integrals agree. The published continuity estimate makes every such regular functional a distribution; the published embedding theorem supplies the converse implication $T_u=T_{\tilde{u}}\Rightarrow u=\tilde{u}$ almost everywhere under Countable Choice.