TITLE: Existence of independent and identically distributed random variables.
QUESTION [27 upvotes]: I often see the sentence "let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with a certain distribution". But given a random variable $X$ on a probability space $\Omega$, how do I know that there is a sequence of INDEPENDENT random variables of the same distribution on $\Omega$?

REPLY [7 votes]: A very concrete approach to construct a sequence of iid random variables with distribution $F$ (say, $F$ is the desired cumulative distribution function) is to proceed as follows.
It is enough to construct a sequence of iid uniform $[0,1]$ random variables $(X_i)_{i\ge 1}$, because then $(F^{\leftarrow}(X_i) )_{i\ge 1}$ is a sequence of iid random variables with common distribution $F$. Here, $F^\leftarrow$ is the generalized inverse of $F$.
So let's concretely construct a sequence of iid uniform $[0,1]$ random variable.
Take $\Omega = [0,1]$ and $\mathbf P$ be the Lebesgue measure on $\Omega$; there we naturally have a single measurable function (or random variable) $U:\Omega\to\mathbb R$ defined by $U(x)=x$ for any $x\in \Omega$.
Now view $U$ as a random variable, and write its decomposition in base $2$ as
$$U = \sum_{k=1}^{+\infty} B_k/2^k.$$
Exercise: show that each $(B_k)_{k\ge 1}$ is a sequence of iid Bernoulli$(1/2)$ random variables on $\Omega$.
Then we may split the set of positive integers into countably many infinite subsets; for instance, let $(p_n)_{n\ge 1}$ be the sequence of prime numbers and fr any $n\ge 1$, let $I_n = \{ p_n^j, j=1,2,3,4,... \}$. Each $I_n$ is infinite and $I_n\cap I_m = \emptyset$ for any $n\ne m$. From these sets $I_n$, for any $n$ define
$$X_n = \sum_{k=1}^\infty B_{\varphi_n(k)}/2^k$$
where $\varphi_n$ is a bijection $\mathbb N_{\ge 1}\to I_n$ (that is, the countable set $I_n$ can be written as $I_n = \{\varphi(1), \varphi(2),\varphi(3), ...\}$. Because the sets $I_n$ are disjoint, a Bernoulli random variable $B_k$ is used only in one of the $X_n$'s, so we can prove the following to conclude:
Exercise 2: show that each $X_n$ has the uniform distribution on $[0,1]$, and that the sequence $(X_1,...,x_n,...,)$ is iid.
Hence $\Omega=[0,1]$ equipped with the Lebesgue measure is rich enough to construct a sequence of iid random variables with distribution $F$.