Probability density function: what's a measure $X_*P$? The Wikipedia article on PDFs defines the probability distribution of a random variable $X$ in $(\mathcal{X},\mathcal{A})$ to be the measure $X_*P$ on $(\mathcal{X},\mathcal{A})$. I understand what a measure is, but what does the notation $X_*P$ mean?
Also, the article then gives the equality
$\int\limits_{X^{-1}A}dP=\int\limits_{A}fd\mu$
where 
$f=\frac{dX_*P}{d\mu}$.
How did $dX_*P$ become $dP$, what what's the relation between $X_*$ and $X^{-1}$? What does it even mean to take the inverse of a variable?
 A: A random variable $X \colon \Omega \to \mathcal{X}$ is a measurable function, where $(\Omega, \mathcal{F}, P)$ is a probability space and $(\mathcal{X}, \mathcal{A})$ is a measurable space. So $X^{-1}A = \{\omega \in \Omega \colon X(\omega) \in A\}$ is just the inverse image of $A$ under $X$.
The measure $X_*P$ is defined on $(\mathcal{X}, \mathcal{A})$ via $$X_∗P(A)=P(X^{−1}(A)).$$ It is called a probability distribution of $X$. Informally, the probability distribution is used to answer the questions like "what is the probability of $X$ falls within a particular range of values?".
Also $dX_*P$ didn't become $dP$, $\frac{dX_*P}{d\mu}$ is a solid symbol that denotes the Radon-Nikodym derivative of the measure $X_*P$ w.r.t. the measure $\mu$. The equality holds because of the definition of that derivative, the definition of $X_*P(A)$ and the equality $P(F) = \int_{F}dP$ for any measurable $F \in \mathcal{F}$. This derivative is called a probability density function of $X$. If $\mu$ is the Lebesgue measure, then $f$ is just usual probability density function for continuous random variables. For discrete case, $\mu$ is the counting measure and $f$ is called a probability mass function.
When you're considering some random variable, ussually you're not interested in the particular form of the mapping $X \colon \Omega \to \mathcal{X}$ but you're interested in its distribution, that can be specified using a probability density function.
