What is the difference between a probability measure and a probability density function? [duplicate]

Question

During my research, I have repeatedly come across the terms probability measure and probability density function (pdf). I am familiar with the concept of a pdf, but I am not entirely sure how probability measures relate to pdfs, or where exactly the differences lie in theoretical terms. I have since combed StackExchange (such as this and this question), and visited the Wikipedia article for probability measures, but I still cannot see the difference, so I imagine I must be missing a nuance.

Hence my question: Could you check my understanding, and - if necessary - correct it?

To provide an example for context, let us consider the continuous-valued example of human body height. Let us denote the height of a person in meters as a random variable $X$, which maps a person's physical height to a corresponding real value $x$. In that case:

• we could define a probability density function $p(x)$ over all theoretically possible heights $x$. The pdf must be non-negative for all $x$, i.e. $p(x)\geq0 \; \forall \; x$, and it must integrate to $1$, i.e. $\int_{-\infty}^{\infty} p(x) dx=1$. We can integrate $p(x)$ over subintervals to learn the probability that a person's height is in this interval.

• the definition of a probability measure seems related. Let us denote the set of all possible body heights $x$ as $S_1$. Based on this set, we design another set $S_2$ which contains $S_1$ and all possible subsets of $S_1$. The probability measure then assigns values between $0$ and $1$ to each entry in $S_2$, with the condition that the entry $\{\text{all }S_1\}$ in $S_2$ is assigned $1$, $\{\}$ is assigned $0$, and certain additivity properties are met (see the image below, taken from Wikiedpa)

Concerning my understanding of the difference: A pdf $p(x)$ can assign values $\geq1$ to $x$, as it describes a density and only requires that the integral over all $x$ be equal to $1$. The limitation of probability measures to values below $1$ would probably correspond to a role more similar to integrating (vanishingly small) sub-intervals of $p(x)$. Is this the distinction between pdfs and probability measures? Why is this distinction important? Am I missing or misunderstanding something?

Answer 1

A probability density function induces a probability measure with Reimann-Stiljes integration, but the space of probability measures is bigger. Take, as an example, the Cantor distribution which has no probability density function, but whose distribution function, the Cantor function, meets criteria of a probability measure, (i.e. it is cadlag, tends to 0 as $x \rightarrow -\infty$ and tends to 1 as $x \rightarrow \infty$). The Cantor function is an intuitive and elementary example in measure theory of a function that is continuous measurably everywhere, has derivative of 0 measurably everywhere, but is not constant.