Is there some way of describing the co-domain of probability density functions? Does it relate in some way to something physically meaningful? I was given that question today - and I was at a loss. Density for me, is the co-domain of pdfs - a scalar dimension with values from zero to infinity.
For instance, I fit a normal distribution to a probability vector on N values (a probability mass function). This vector contains normalised values of the actual frequency counts in a histogram with N bins. The normalisation is done by dividing each frequency count by the discrete integral "area" of the pmf - so that the pmf values sum to 1. The pmfs now seems to be scaled similarly to the estimated pdfs.
Is probability density a measure of probability? I am sure that the necessary definitions must be hidden somewhere deep down in the guts of measure theory - which is why I have included that as a tag.