I'm currently reading a book on measure-theoretic probability theory, and I'm having trouble seeing how the familiar objects distributions, pdfs/pmfs, and random variables from my calc-based prob/stat classes fit into this new language:
$$\begin{array} A(\Omega,\scr H,\mathbb{P}) & \stackrel{X}{\longrightarrow} & (E,\scr E,\mu) \\ \ & & \downarrow{\hat{\mathbb{P}}:= \mathbb{P}\circ X^{-1}} \\ & \ & ([0,1],\scr B([0,1])) \end{array} $$
Let $\mu$ be a measure on the measurable space $(E,\scr E)$. $\hat{\mathbb{P}}=\mathbb{P}\circ X^{-1}$ is the distribution of the random variable $X$, and it too is a measure on $(E,\scr E)$. The probability density function $f_X$ of $X$ is the Radon-Nikodym derivative of $\hat{\mathbb{P}}$ relative to $\mu$. Then $\hat{\mathbb{P}}(A):=\mathbb{P}(X^{-1}(A))=\int_A d(\mathbb{P}\circ X^{-1})=\int_Ad\mu f_X$ defines a probability measure on $(E,\scr E)$. I'm a little confused because I see that in many sources that $f_X$ is the distribution of $X$, not the density, so I'm not sure if it's a notational thing or I'm not understanding things.
The two examples I saw in my calc-based prob/stat classes were
Discrete: In this case, the image of $X$ is countable. $(E,\scr E)$ is usually $(N,2^N)$, $N\subseteq\mathbb{Z}$. Since every measure $\mu$ on the discrete space $(N,2^N)$ is discrete, $\mu(A)=\sum_{x\in E}m(x)\delta_x(A)$ (which implies $\int_A d\mu f=\sum_{x\in A}m(x)f(x)$=$\sum_{x\in A}\mu(x)f(x)$), the distribution $\mathbb{P}\circ X^{-1}$ has this form. Then since singletons are measurable sets, $\hat{\mathbb{P}}(\{x_0\})=\int_{x_0}d\mu f_X=\mu(x_0)f_X(x_0)$. Here I'm a litte confused about the presence of the $\mu(x_0)$ term, since We need $\hat{\mathbb{P}}(\{x_0\})=f_X(x_0).$
Continuous: In this case, the image of $X$ is uncountable. $(E,\scr E,\mu)$ is usually $(\mathbb{R}^n,\scr B(\mathbb{R}^n),\lambda)$, with the Lebesgue measure $\lambda$ defined on it.
If anyone could point to any misunderstandings and correct them, it would be greatly appreciated.
You can't choose the distribution $\mu$ on $(E, \mathcal{E})$ arbitrarily, the measure $\mathbb{P} \circ X^{-1}$ needs to be absolutely continuous with respect to $\mu$. Else the density $f_X$ doesn't exist.
Your understanding of the distribution is correct. Whoever calls $f_X$ the distribution and not the density is wrong. Note however, that the density [if it exists] characterises the distribution completely. So when an exercise asks for the distribution of a random varibale, this distribution may be characterised through its density.
The calculation for the discrete case is correct. Note that usually you take the measure $\mu$ to be the counting measure, so $\mu(\{x_0\}) = 1$ and you get the usual definition of the PMF.