Locally integrable functions embed in distributions

F1 and F2 show $u_f$ is a distribution, with compactwise order at most zero; linearity of the integral gives linearity in the equivalence class of $f$. For a bounded test set $B$, take its common compact $K$ and finite $M_B={\sup }_{\phi \in B}{p}_0(\phi )$. Then $p_B(u_f)\le M_B{\int }_K\mathrm{\mid }f\mathrm{\mid }$. Thus each strong seminorm of the image is bounded by a constant times a defining local $L^1$ seminorm, proving continuity, including for nets. If $M_B=0$, the left side is zero.

Suppose $u_f=0$, and fix a closed ball $H$ compactly inside $\mathrm{\Omega }$. Take $\chi$ from F5 equal to one near $H$ and compactly supported in $\mathrm{\Omega }$, and extend $g=\chi f$ by zero to ${\mathbb{R}}^n$; it is in $L^1$ by local integrability. Also use F5 to obtain a nonnegative smooth compact bump equal to one on a ball, and divide it by its positive finite integral to obtain $\rho$ of mass one. Fix $R$ containing its support. For sufficiently small $\epsilon$, every $x\in H$ has $x-\mathrm{supp}{\rho }_{\epsilon }$ inside the neighborhood where $\chi =1$: the compact $H$ has positive distance from that neighborhood's closed complement. Consequently [step 1.1, given, F3, F5] $g\ast {\rho }_{\epsilon }(x)={\int }_{\mathrm{\Omega }}f(y){\rho }_{\epsilon }(x-y)dy=u_f({\rho }_{\epsilon }(x-\cdot ))=0.$ The integrals are absolutely finite because $g\in L^1$ and the kernel is bounded; the reflected kernel is a test compactly supported in $\mathrm{\Omega }$.

By F3, F4 and F8, $\mathrm{\parallel }g\ast {\rho }_{\epsilon }-g{\mathrm{\parallel }}_1\to 0$. Since $g=f$ on $H$ and step 2.1 makes the convolution zero there, ${\int }_H\mathrm{\mid }f\mathrm{\mid }\le \mathrm{\parallel }g\ast {\rho }_{\epsilon }-g{\mathrm{\parallel }}_1\to 0$. Hence $f=0$ almost everywhere on $H$. The closed rational balls compactly inside $\mathrm{\Omega }$ are countable and their interiors cover $\mathrm{\Omega }$; applying this conclusion to each and taking the countable union of the null sets $\{x\in H:f(x)\mathrm{\ne }0\}$ gives $f=0$ almost everywhere on $\mathrm{\Omega }$. No family of cutoffs was selected simultaneously. Apply the same argument to $f-h$ for injectivity of the map on equivalence classes.

As a useful convergence consequence, if $f_j\to f$ almost everywhere and for every compact $K$ there is an integrable majorant of all $\mathrm{\mid }{f}_j\mathrm{\mid }$ on $K$, F7 gives ${\int }_K\mathrm{\mid }{f}_j-f\mathrm{\mid }\to 0$. Step 1.1 then gives strong convergence $u_{f_j}\to u_f$. On the empty domain there is only the zero class and zero distribution, so injection and continuity still hold. $\mathrm{\square }$