I’m studying mathematical statistics.
I learned about the concepts of sample space, $\sigma$-field, probability set function with probability axioms, random variables, and probability density function.
Let $S$ be a sample space. Let $A$ be a sigma field. Let $\mathbb{P}$ be a probability set function on $A$. And let $X$ be a random variable on $S$.
Then we have a new sample space $X(S)$. And its power set is a new sigma field $\mathcal{F}$. So, now we can define a function $\mathbb{P}_X$ from $\mathcal{F}$ to $\Bbb{R}$ by $\mathbb{P}_X(B) = \mathbb{P}_X[X^{-1} (B)]$ for all $B \in \mathcal{F}$.
In my book, the probability density function is defined in that way. And it says this probability density function $\mathbb{P}_X$ is always a probability set function. The book says it is an exercise. But I don’t agree with this. I think probability density function is a probability set function only if it is well-defined!!
In some cases, it might be possible that $X^{-1} (B)$ is not in $A$, which means $\mathbb{P}_X$ is not well-defined since $\mathbb{P}[X^-1(B)]$ is not well-defined.
So I think $\mathbb{P}_X$ is not always well-defined.
1)
You are speaking of a new sample space $X(S)$, but usually not the image of $X$ is used for that but the codomain $\mathbb R$ of function $X$.
2)
The powerset of $\mathbb R$ is indeed a sigma field, but is not used in the new probability space. Practicized is probability space $(\mathbb R,\mathcal B,\sf P_X)$ where $\mathcal B$ denotes the sigma field of Borel sets. $X$ being a random variable means that $X^{-1}(B)$ is an element of sigma field $\sf F$ for every Borel set $B$, so $\sf P_X(B)$ is well defined for every $B\in\mathcal B$.
3)
If in your book $\sf P_X$ is named a "probability set function" then that is the naming of what I would call a "probability measure". I cannot imagine that they would mean "probability density function" which is another concept.