Hmm, I'd imagine the RTR is a typo because typically you don't want to scale and you want to be normalized to 1 when you integrate over the domain -\inf to \inf. That's what the term out in the front is for. I don't have the book on me at the moment to see in context what they are talking about.
In other words: we are compensating physically implausible assumptions (perfect pinnhole) with more physically nonsense (infinetly high sensor sensibility) which still leads to implausible results (infinetly high measurements in the case of perfect reflections).
They aren't implausible assumptions otherwise we wouldn't make them to begin with! Hopefully,no infinities creep in to our answer otherwise we we would get flux of infinite values and that's not very useful.
Our end goal is to obtain plausible results that are close to the ground truth. The only reason we use simplifying assumptions and simpler models is that we want simpler calculations. We know that with our simplified models we get results that are very close to the ground truth.
An example is to look at the general form of the radiation transfer equation (it describes how much flux is transfered between two surface elements). The general form requires double integrals over both surfaces which is computationally expensive. Sometimes its good enough to say that we can approximate the result as two point sources and ask the question how much energy is transfered between two point sources. These approximations will be valid and come to the right solution, for our purposes, if the two points are far enough away and the surfaces are small enough.
Since the point-to-point radiation transfer equation gave us the same answer, within our acceptable tolerance, and with no integrals we are happy to use it. Additionally, with some mathematical foot work you can show that the point-to-point transfer equation is derived directly from the defitions of irradiance, intensity and solid angle so its mathematically sound with its feet firmly planted in the ground of physical plausability.
In the same vein, it's ok to use a pinhole model and if you do it correctly and you make some assumptions about your scene, light and camera then the result should be very similiar to if you had an aperature, integrated over sensor area, over time and all wavelengths.
For example, you could write a simple raytracer that did a montecarlo integration at every pixel with a very small aperature for one spherical area light very far away from the scene and it would come very close to a rasterizer that used the point light source and a pinhole camera.
The above should be a little more clear. We want to develop a BRDF that for the intensity of the light coming in one direction, \L_i, should be equal to the light going out in the direction of reflection.
In the final BRDF, should the second delta function be [link]? The sign of pi doesn't matter, if I'm thinking about this correctly.
Yah, sorry you want to setup the dirac delta function so that the angle away from the normal axis is equivalent and the other angle is equivalent but rotated 180 degrees.
Then I think of how a film camera works, you have a photographic film that gets darker when photons hit it, thus making the final image brighter (because the film is a negative)
Right, as you point out, you will want to integrate over the visible hemisphere(aperture), sensor area, and time (equivalent to shutter speed) which beyond just controlling exposure will give you phenomenon such as motion blur and depth of field. If instead of integrating RGB you additionally integrate over the wavelengths then you can get other phenomenon such as chromatic aberration.