Difference in Probability Measure vs. Probability Distribution I am trying to better understand the Difference in "Probability Measure" and "Probability Distribution"
I came across the following link : https://math.stackexchange.com/questions/1073744/distinguishing-probability-measure-function-and-distribution
" The difference between the terms "probability measure" and "probability distribution" is in some ways more of a difference in connotation of the terms rather than a difference between the things that the terms refer to. It's more about the way the terms are used. "
The answer over here suggests that these two concepts might be the same thing?
In this case - could we consider the "Normal Probability Distribution Function" as a "Probability Measure"?
Thanks!

First off, I am not used to the term "Probability Distribution Function". In case you are referring to a "PDF" when you are using "Probability Distribution Function", then I would like to point out that PDF is rather the abbreviation for "Probability Density Function". Below, I will presume you meant probability density function.
Second, the word "distribution" is used very differently by different people, but I will refer to the definition of this notion in the scientific community.
In a nutshell: The distribution of a random variable $X$ is a measure on $\mathbb{R}$, while the PDF of $X$ is a function on $\mathbb{R}$ and the PDF doesn't even always exist. So they are very different.
And now we get to the mathematical details:
First, let's define the term random variable because that is what all those terms usually refer to (unless you get a step further and want to talk about random vectors or random elements, but I will restrict this here to random variables). I.e., you talk about the distribution of a random variable.
Given a probability space $(\Omega, \cal{F}, p)$ ($\Omega$ is just a set, $\cal{F}$ is a sigma algebra on $\Omega$, and $p$ is a measure on $(\Omega, \cal{F})$), a random variable $X$ is a measureable map $X: \Omega \to \mathbb{R}$.
Then we can define: The distribution $p_X$ of the random variable $X$ is the measure $p_X = p \circ X^{-1}$ on $\mathbb{R}.$
I.e. you push the measure $p$ from $\Omega$ forward to $\mathbb{R}$ via the measurable function $X$.
Next we define the probability density function (PDF) of a random variable $X$: The PDF $f_X$ of a random variable $X$, if it exists, is the Radon-Nikodym derivative of its distribution w.r.t. the Lebesgue measure $\lambda$, i.e. $f_X = \frac{d\,p_X}{d\,\lambda}$.
So the distribution $p_X$ and the PDF $f_X$ of a random variable are very different entities (to a stickler, at least). But very often, the PDF $f_X$ exists, and then it contains all of the relevant information about the distribution $p_X$ and can thus be used as a handy substitute for the unwieldy $p_X$.
