Let $X_1$ be given as a random variable on a probability space $(\Omega, \mathcal{F}, P)$. Can we always guarantee the existence of i.i.d (independent and identically distributed) random variables $X_2$, $X_3$, $\cdots$ on the same space? If so, how...?
Any help would be very much appreciated!
No. Consider for example $\Omega=\{0,1\}^{3}$, $\mathcal{F}=2^\Omega$, and $$\mathbb{P}(A)=\frac{|A|}{|\Omega|}.$$ Suppose $X_i\sim\text{Bernoulli}(1/2)$ for $i=1,2,\dots,n$ are i.i.d. Then $$\mathbb{P}(X_1=0,\dots,X_n=0)=\prod_{i=1}^n\mathbb{P}(X_i=0)=1/2^n.$$ However, we also have $$|\Omega|\cdot \mathbb{P}(X_1=0,\dots,X_n=0)=|\{X_1=0,\dots,X_n=0\}|\in\mathbb{Z}.$$ Thus $|\Omega|/2^n=1/2^{n-3}\in\mathbb{Z}$ so $n\le 3$. Thus given $X_1\sim \text{Bernoulli}(1/2)$ in this case, we can only construct at most $3$ i.i.d. variables.