INTRODUCTION. Let's consider some previous answers (from previous questions) about "probability measure" and "probablity distributions":
(Previous question 1) Distinguishing probability measure, function and distribution,
(Previous answer 1.1) Michael Hardy wrote:
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. A probability distribution or a probability measure is a function assigning probabilities to measurable subsets of some set.
(Previous answer 1.2) While Maverick Meerkat wrote:
I think "Probability Distribution" might mean either of these things, but I think most often it will be used in less mathematically precise as it's sort of an umbrella term - it can refer to the distribution of measure on the original space, or the distribution of measure on the real line, characterized by the CDF or PDF/PMF.
Usually, if there's no need to go deep into the math, people will stay on the level of "probability function" or "probability distribution". Though some will venture to the realms of "probability measure" without real justification except the need to be absolutely mathematically precise.
(Previous question 2) Probability measure and distribution
(Previous answer 2.1) ncmathsadist wrote:
Every right continuous nondecreasing function $:ℝ→[0,1]$ with $(∞)=1$ and $(−∞)=0$ is a probability distribution, where $_()=_((−∞,]),∈ℝ$ and $_$ is a Borel measure on $ℝ$.
SUMMARY OF THE INTRODUCTION. To my understanding, it looks like tha we can call "probability measures" as "probability distributions", and viceversa. It looks like that "probability distributions" is a shallow term, while "probability measures" is a deeper term, used in probability theory. Right or wrong?
MY QUESTION
I am currently using a number of measures of distances/divergences among pairs of "probability distributions" $P$ and $Q$, and I would like to use also the Wasserstein distance $W_1$. However, $W_1$ is defined for "probability measures", $\mu$ and $\nu$, i.e. $W_1(\mu,\nu)$. Therefore, can I still call the two "probability measures", $\mu$ and $\nu$, as "probability distributions", like $P$ and $Q$?
