Why is the Radon–Nikodym theorem important in probability? I was reading this Wikipedia page and came across the following statement:

The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν can be expressed in this way with respect to another measure μ on the same space. The function  f  is then called the Radon–Nikodym derivative and is denoted by {\displaystyle {\tfrac {d\nu }{d\mu }}}{\displaystyle {\tfrac {d\nu }{d\mu }}}.1 An important application is in probability theory, leading to the probability density function of a random variable.

I am trying to understand why this important - i.e., why is the Radon–Nikodym theorem important in defining the probability density function of a random variable?
In the courses I have taken in statistics/probability (engineering), we were always shown the definition of a probability distribution (of a random variable) without any mention of the Radon-Nikodym theorem. I would have never even known that such a theorem existed, let alone that such a theorem would be so important in defining the probability density of a random variable.
Why is the Radon-Nikodym theorem so important in defining the probability density of a random variable? Is it really that important that it can not be defined without this theorem?
 A: The theorem isn't necessary for defining the density of a random variable. After all, any measurable nonnegative function that integrates to $1$ is a density. One major application of the Radon-Nikodym theorem is to prove the existence of the conditional expectation. Really, the existence of conditional expectation is very much related to the Radon Nikodym theorem, and in fact can be used to prove the theorem. See https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf for details.
A: I think, given the background that you mentioned that you have, understanding the Radon-Nikodym Theorem is not necessarily going to bring you any clarity in it's connection with Probability. But you can think about it as an analogue of the First Fundamental Theorem of Calculus.

Suppose that we have $f(x)$, a continuous function on a closed interval $[a,b]$. Then we can construct an anti-derivative for $f(x)$ via
$$F(x) = \int_a^x f(t) \; dt$$ for all values of $x \in [a,b]$.

You can think about "measure" as a fancy mathematical term for "length." So given any two points $x \leq y$ in $[a,b]$ you can determine the length of the interval between those two values as $y-x$. This new function $F(x)$ gives you a way of defining another length on $[a,b]$. You could calculate another distance between those two points as $$F(y) - F(x).$$
If you get down into the gritty details, this new "measure" of length has many of the same properties as the old "measure." But it tells us something different, it can tell us about the probability of an event occurring. The Radon-Nikodym theorem just states that we can express the second measure using the first (but there are many mathematical details that are being brushed under the rug in that explanation).
DISCLAIMER: I am not an analyst and have most likely forgotten many key details about the Radon-Nykodym Theorem, but I hope this helped at least a bit.
