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!

**Answer**

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 R, while the PDF of X is a **function** on 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 (Ω,F,p) (Ω is just a set, F is a sigma algebra on Ω, and p is a measure on (Ω,F)), a *random variable* X is a **measureable** map X:Ω→R.

Then we can define: *The distribution pX of the random variable X is the measure pX=p∘X−1 on R.*

I.e. you push the measure p from Ω forward to R via the measurable function X.

Next we define the *probability density function* (PDF) of a random variable X: *The PDF fX of a random variable X, if it exists, is the Radon-Nikodym derivative of its distribution w.r.t. the Lebesgue measure λ, i.e. fX=dpXdλ.*

So the distribution pX and the PDF fX of a random variable are very different entities (to a stickler, at least). But very often, if the PDF fX exists, it contains all of the relevant information about the distribution pX and can thus be used as a handy substitute for the unwieldy pX.

**Attribution***Source : Link , Question Author : stats_noob , Answer Author : frank*