Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation random variables $X_n$ defined below do the job:
$$X_n((t_m)_{m=1}^\infty) = t_n\qquad ((t_m)_{m=1}^\infty\in \{0,1\}^{\mathbb{N}},\,n\in \mathbb{N}).$$
According to Kchinchine's inequality the linear span of $\{X_n\colon n\in \mathbb{N}\}$ in $L_1(\mu)$ is isomorphic to $\ell_2$.
Are there easy ways to construct sequences of independent random variables that span copies of $\ell_p$ for $p\in (1,2)$ in $L_1(\mu)$?
Such sequences exist (for instance if you take a sequence of independent $p$-stable random variables, then the linear span will be even isometric to $\ell_p$), however the formulae for such random variables will be rather complicated. In other words, I put emphasis on a nice closed-form formula for $X_n$ rather than the very existence.