Consider X as a continuous random variable which can assume any value in [0, 1]. It is known that P(X=x)=0 where P is the probability density function. I want to understand this intuitively.
The math insight article helps me somewhat:
In other words, the probability that the random number X is any particular number x∈[0,1] (confused?) should be some constant value; let's use c to denote this probability of any single number. But, now we run into trouble due to the fact that there are an infinite number of possibilities. If each possibility has the same probability c and the probabilities must add up to 1 and there are an infinite number of possibilities, what could the individual probability c possibly be? If c were any finite number greater than zero, once we add up an infinite number of the c's, we must get to infinity, which is definitely larger than the required sum of 1. In order to prevent the sum from blowing up to infinity, we must have c be infinitesimally small, i.e., we must insist that c=0.
But, what if I choose, c=1/N (where N is that large number) for the uniform case? The sum will still be 1 as far as I can understand. For the non-uniform case, I can pick some 0's and others non-zeros and still be theoretically able to get a sum of 1 for all the possible values.
Anything that helps me understand this clearly will be of immense help. The answer here helps but I still don't get it.