I know that not every probability measure admits a probability density function, where a function is a mapping $f: X \to \mathbb R$.
However, there is something, such as the Dirac delta distribution, which is not a function, because it is defined as that whose integral is the step function. There is, of course, no mapping $f:X\to \mathbb R$ that has as its integral the step function, so the Dirac delta distribution is not a function. (It is a "generalized function").
Is it possible for every conceivable (probability) measure on every conceivable space, to define for it a (probability)-measure density distribution (i.e. NOT necessarily a function), using tools similar to the Dirac delta distribution?