# Correspondence between the "density function of a probability measure" and the "probability density function" (PDF)

Question. If there is a one to one correspondence between a "borel probability measure" on the line $$\mathbb{R}$$ and a "cumulative distribution function" (CDF) (please see on page 13, "Theorem 1.2. (Lebesgue)", in S.R.S.Varadhan(2001), Probability Theory), is there, as well, a one to one correspondence between the "density function of a probability measure" and the "probability density function" (PDF)?

Indeed, on page 384, in Levin & Peres (2017) Markov Chains and Mixing Times, second edition, we can read as follows:

Given a density function $$f$$, the set function defined for Borel sets $$B$$ by $$$$\mu_f(B) = \int_{B} f(x)dx$$$$ is a probability measure.

• Please tell us how, if at all, the mathematical definitions of "density function of a probability measure" and "PDF" differ.
– whuber
Commented Sep 20, 2023 at 19:52
• From your comment I deduce that the answer is: "yes, there is a one to correspondence between...", thanks
– Ommo
Commented Sep 20, 2023 at 20:15

## 1 Answer

The probability density function (PDF) is indeed referring to the density of a probability measure, so these are the same thing. You should bear in mind that the density is formally defined as the Radon-Nikodym derivative of the probability measure with respect to a dominating measure. There can be different densities (i.e., different PDFs) corresponding to different dominating measures, though we generally use the same standard dominating measures for discrete and continuous random variables (counting measure for discrete random variables and Lebesgue measure for continuous random variables). You should also bear in mind that the density can be varied on a set with zero probability measure without becoming invalid, so the density is generally non-unique for continuous random variables. (We can sometimes establish uniqueness by adding certain continuity requirements, etc.).

• Thanks a lot @Ben!
– Ommo
Commented Sep 21, 2023 at 7:02