probability measures vs. probability distributions vs. measure of probability density I am learning probability theory right now and am confused about some basic concepts. I have a few questions and am wondering if you can also check if the following is correct:
Suppose we have a probability space $(\Omega, \mathcal{F}, \mathbb{P}^1)$. My understanding is that:


*

*$\mathbb{P}^1$ is called a probability measure but not a probability distribution.

*If we have some random variable $X$ that maps to $(\mathbb{R}, \mathcal{B})$, then $X$ induces a probability distribution $\mathbb{P}^2$ on $(\mathbb{R}, \mathcal{B})$, which is a measure on $(\mathbb{R}, \mathcal{B})$ such that $\mathbb{P}^2(A) = \mathbb{P}^1(X^{-1}(A)), A \in \mathcal{B}$. Is it true that $\mathbb{P}^2$ is also a probability measure on $(\mathbb{R}, \mathcal{B})$?

*We can consider a probability density function $f$ of $X$ with respect to some dominating measure $\mathbb{M}$ on $(\Omega, \mathcal{F})$. Then $\mathbb{F}(A)=\int_A f ~d\mathbb{M}, A \in \mathcal{F}$ is a measure on $(\Omega, \mathcal{F})$. Is it always true that $\mathbb{F}$ is a probability measure on $(\Omega, \mathcal{F})$?

*Must the random variable $X$ be defined on a probability space? From the above point, people often take the dominating measure $\mathbb{M}$ to be the Lebesgue measure. But the Lebesgue measure is not a probability measure...

 A: Hope I'm not too late :)

*

*A probability distribution is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, or more generally on a metric space, which is induced by a random variable or random element. By induced, I mean given a random variable $X : \Omega\to \mathbb{R}$, the probability distribution of $X$ is the set function $$P_X : \mathcal{B}(\mathbb{R})\to[0,1], P_X(A)=\mathbb{P}(X^{-1}(A))$$
Every probability measure on the real line (or a metric space) can be shown to be the probability distribution of some random variable (or random element). However, it is a convention that we only call a probability measure a probability distribution if it is the probability measure induced by a specified random variable (or element). It is only a convention though.


*That is true. You can show that the three axioms of probability are satisfied. It comes from the fact that the inverse image works very nicely with set operations.


*Yes. I'm assuming you mean that we have a probability measure $\mathbb{P}$ on $(\Omega,\mathcal{F})$ as well as some dominating measure $\mathbb{M}$ on there. If $f$ is the density of $X$ then by definition of the density, $\forall A\in\mathcal{B}(\mathbb{R})$,
$$P_X(A)=\mathbb{P}(X\in A)=\int_A f(x)\, d\mathbb{M}(x)$$
That is, $\mathbb{F}=P_X$ and so if you believe that $P_X$ is a probability measure, then so is $\mathbb{F}$.


*Sort of no, not really. Things get a little confusing here. Given a measurable space $(\Omega,\mathcal{F})$, any function $f : \Omega\to \mathbb{R}$ which is measurable is called a measurable function. In the case that we have a measure space $(\Omega,\mathcal{F},\mu)$, $f$ is still called a measurable function. In the special case that $\mu$ is a probability measure and we are working in probability theory then $X$ is conventionally used instead of $f$ and we call $X$ a random variable. So to answer your question, a random variable can be defined on any measurable space, not just probability space. But usually when you use the term 'random variable', it is implied that we are working with a probability space, in probability theory. In addition, be careful that the probability space on which a random variable is defined usually does not have the Lebesgue measure on there (unless you're working with say $[0, 1]$).
