Is it possible to define distributions acting on non-smooth functions?
The reason I'm asking is because of the rendering equation $$L_o(x,\omega_o) = \int_\Omega f_r(x,\omega_o,\omega_i) L_i(x,\omega_i) \text{d}\omega_i.$$
In this equation, $f_r$ is the Bidirectional Reflection Distribution Function (BRDF). However, a common BRDF is the Dirac delta distribution (for optically perfect mirrors). This makes me want to define $f_r$ as a distribution. The rendering equation would then look like $$ L_o(x,\omega_o) = \langle f_r(x,\omega_o),L_i(x) \rangle$$ which I think seems nice.
The problem is that since $L_i$ is the incoming light at $x$ from direction $\omega_i$, it is often a non-smooth function. According to the little I know about distributions, they only act on test functions in $C^\infty$ (and with compact support, which isn't a problem). Does this mean that I cannot define the BRDF as a distribution?
What would then be a mathematically rigorous way of stating the rendering equation while allowing $f_r$ to be the Dirac delta distribution?
Update:
I think I found what I was looking for: Mollifiers. I should be able to convolve $L_i$ with a Mollifier: $L_{i,\epsilon} = \phi_\epsilon \ast L_i$. This gives me a smooth function and I can then define the rendering equation as $$ L_o(x,\omega_o) = \lim_{\epsilon \to 0} ~\langle f_r(x,\omega_o),L_{i,\epsilon}(x) \rangle$$ Does this look correct?