Random variable exponentially distributed? I just want to be sure about this:
If I read the phrase ' a random variable is exponentially distributed'( which is often said in probability theory and then it is never explictely stated what $X$ actually is and it is just looked at the probability itself) this could  rigorously mean, that there is a map $X: \Omega \rightarrow \mathbb{R}$ and that there is a probability space $(\Omega = \mathbb{R}_{>0}, X^{-1}(\mathscr{B}) , P)$ (where $X^{-1}(\mathscr{B})$ is the inverse map of the sigma algebra of the Borel sets) such that $P(\{X^{-1}(E)\}):= \int_\mathbb{R} \chi_E(x) \lambda e^{-\lambda x}dx$?
This is one interpretation that came to my mind by thinking about what it could mean. Could anybody here now explain to me where I am wrong and what it actually means? ( So my goal is to give all the details to the phrase ' a random variable $X$ is exponentially distributed)
Does anybody here have an idea? If there is anything unclear about my question, please let me know.
 A: A real valued random variable $X$ is exponentially distributed with parameter $\lambda$ if and only if its cumulative distribution function (CDF) is $$F(x)=P(X \leq x) = \left\{\begin{array}{lr}1- e^{-\lambda x}&,x\geq 0\\0&,o.w.\end{array}\right. $$
Note that having a particular distribution (in this case, exponential) doesn't imply $X$ lives on any particular probability space. It does, however, specify $P(X \in B)$ where $B$ is a Borel-measurable subset of $\mathbb{R}$. 
Given a CDF  $F$, you can easily construct an appropriate probability space for which a random variable with that CDF lives on (this is easy to do - get the probability space $(\Omega,\mathcal{F},P)$ with $\Omega = (0,1)$, $\mathcal{F}$ the Borel sigma algebra on $\Omega$ and $P$ be the lebesgue measure, and let $X(\omega)=\sup\{ y : F(y) < \omega \}$ is probably the simplest construction of a probability space and random variable with distribution $F$). However, you can come up with infinitely many such spaces and random variables (with the same distribution), which can look very different.
Having a particular distribution is not a property of any particular underlying probability space - this is why you can have sequences of random variables which live on different probability spaces "converge in distribution" in a meaningful way!
