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!

You are right, when we are just starting out our stat studies, these terms can be super confusing, especially to people that are very detail-driven or very particular about terminologies. You do need a very good understanding here before you can master the applications of them. And using R functions such as pnorm(), dnorm() forces you to understand these.
And I’d say the post you quoted put it pretty well - especially where it pointed out that “probability functions” - “probability density function” and “probability mass function” are precisely defined, and the other terms can be understood as their names suggest - that the “probability measure” is a measure for the probabilities - is it cumulative or not? It doesn’t say. It’s a measure, so it can refer to both. A probability distribution can be similar. But don't confuse it with distribution function. Distribution function in most cases refers to "cumulative distribution function", which is also clearly defined as the probability taking on a value equal to or less than a specified value.
Maybe an example can help

This is a simple normal distribution chart. And here,

*

*The probability function - which is a probability density function  here since it’s continuous variable - describes the red curve. It’s a line, that’s represented by this function
$$\large{f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}}$$
where $\mu$ is the mean and $\sigma$ the standard deviation.


*Calculating integrals under this “red line” gets us different “probability” values - for example, the orange area is an area that represents 1 standard deviation away from the mean with 34% area of the total - which is 34% probability; whereas the blue area represents 17% probability. I’d say the entire chart depicting the probabilities with a normal distribution is a probability measure and a probability distribution. But only and precisely the red line is a probability function, or more specifically, a probability density function.
Don't stress too much about what exactly is the definition for "probability measure"/"probability distribution". I believe as you see more examples, you will feel comfortable using these terms.
