given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.

I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \alpha\circ \sigma^3 \cdots : \Omega \to \Omega$ are all Independent identical distribution :

Let $(\Omega, \mu) = ([0,1], Leb)$, $\alpha $ takes countable $0-1$ value, and $\sigma$ is piecewise linear map and some $r \in (0,1)$ such that:

$$ \alpha(\omega)=\left\{ \begin{aligned} 0 & & \omega \in [0,r] \\ 1 & & \omega \in [r, 1]\\ \end{aligned} \right. $$

$$ \sigma(\omega)=\left\{ \begin{aligned} \frac{\omega}{r} & & \omega \in [0,r] \\ \frac{\omega-r}{1-r}& & \omega \in [r, 1]\\ \end{aligned} \right. $$

We can also construct similar example such that $\alpha$ takes countable values, i.e. distribution of $\alpha$ has only countable atoms.

Can we construct an example on $\Omega= [0,1] $such that

1, $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \alpha\circ \sigma^3 \cdots $ are Independent identical distribution and

2, distribution of $\alpha$ has no atom and absolutely continuous w.r.t Leb? Thanks in advanced!