Are photons countable/uncountable? Lebesgue measure, probability I have a question which is bugging me for a few days now (I'm a math major but I took no physics courses). I am probably not the first person to come up with such question, so I would appreciate any literature suggestions as well.
Are photons countable? Consider the following example:
A flashlight shines on a 1x1 cm paper sheet. It is centered 10 cm above it.
Can we count the amount of photons hitting it? Is the probability of photons hitting any one point equal to zero?  What if they're uncountable?
What if I create an uncountable union of such zero sets? Is it's measure still zero? (probably not, or else it wouldn't be hit at all)
 A: The number of photons that hit the paper during any finite interval of time is not just countable but finite.  This is because each photon has a small but non-zero amount of energy;  specifically, if the wavelength of the light is $\lambda$, then the amount of energy for each photon is $E_0 = hc/\lambda$, where $h$ is Planck's constant and $c$ is the speed of light.
This means that so long as the amount of energy imparted to the paper $E_\text{tot}$ is finite (which it will be for any finite amount of time), then the total number of photons hitting the paper will just be $N = E_\text{tot}/E_0$.  Even if there is a mix of wavelengths, we can still argue that the number of photons will be bounded above so long as the wavelengths are bounded above by some maximum $\lambda_\text{max}$.
If we assume that each photon "hits" the paper at a particular point, then the set of such points is finite and will have Lebesgue measure zero.  That said, the probability density function for a photon to hit a neighborhood of any particular point is not zero, because photons are quantum mechanical objects and have a certain inherent probability "spread" in their locations before they are "measured". (Colliding with a sheet of paper counts as a "measurement" in this context.)
The question about an uncountable union of sets of zero measure is a mathematics question that I don't know how to answer, but honestly it doesn't really seem on topic on a physics forum anyhow.  Best to ask that one over at Math.SE.
