# 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
Sep 20 at 19:52
• From your comment I deduce that the answer is: "yes, there is a one to correspondence between...", thanks
– Ommo
Sep 20 at 20:15