TITLE: Intuition for probability density function as a Radon-Nikodym derivative
QUESTION [44 upvotes]: If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all $x \in \mathbb{R}$.
More rigorously, I could alternatively say that $f$ is the Radon-Nikodym derivative of the distribution measure of $X$ w.r.t. the Lebesgue measure on $\mathbb{R}$, or $f = \frac{\mathrm d \mu_X}{\mathrm d\lambda}$.  As I understand it, $f$ re-weights the values $x \in \mathbb{R}$ in such a way that
$$
\int_B \mathrm d\mu_X = \int_B f\, \mathrm d\lambda
$$
for all Borel sets $B$.  In particular, the graph of $f$ lies below one everywhere:  
so it seems like $f$ is re-weighting each $x \in \mathbb{R}$ to a smaller value, but I don't really have any intuition for this.  I'm seeking more insight into viewing $f$ as a change of measure, rather than a sort of distribution describing how likely $X$ is.  
In addition, does it make sense to ask "which came first?"  The definition for the standard normal pdf as just a function used to compute probabilities, or the pdf as a change of measure?

REPLY [2 votes]: The case you are referring to is valid. In your example, Radon-Nikodym serves as a reweighting of the Lebesgue measure and it turns out that the Radon-Nikodym is the pdf of the given distribution. 
However, Radon-Nikodym is a more general concept. Your example converts Lebesgue measure to a normal probability measure whereas Radon-Nikodym can be used to convert any measure to another measure as long as they meet certain technical conditions. 
A quick recap of the intuition behind measure. A measure is a set function that takes a set as an input and returns a non-negative number as output. 
For example length, volume, weight, and probability are all examples of measures. 
So what if I have one measure that returns length in meters and another measure that returns length in kilometer? A Radon-Nikodym is to convert these two measures. What is the Radon-Nikodym in this case? It is a constant number 1000. 
Similarly, another Radon-Nikodym can be used to convert a measure that returns the weight in kg to another measure that returns the weight in lbs. 
Back to your example, pdf is used to convert a Lebesgue measure to a normal probability measure, but this is just one example usage of measure. 
Starting from a Lebesgue measure, you can define Radon-Nikodym that generates other useful measures (not necessarily probability measure). 
Hope this clarifies it.