Consider a probability space $(X,\Sigma,P)$. Let say that a collection $\mathcal{B}\subseteq\Sigma$ is non-atomic if for every $E\in\mathcal{B}$ and $\alpha\in(0,P(E))$ there exists $F\in\mathcal{B}$ such that $F\subseteq E$ and $P(F)=\alpha$.

Suppose $\Sigma$ non-atomic. Is the $\lambda$-system generated by the collection of all the $2^n$-fold uniform partitions, non-atomic ?

An idea to prove it could be to use transfinite induction since for any collection $\mathcal{B}\subseteq\Sigma$ we have (I think) $\lambda(\mathcal{B})=\cup_{\alpha<\omega_1}\mathcal{B}_\alpha$ with,

- $\mathcal{B}_0:= \mathcal{B}$;
- For every ordinal $\alpha$, $\mathcal{B}_{\alpha+1}:= \{\sqcup_{n\in\mathbb{N}}E_n : \forall n\in\mathbb{N}\ E_n\in \mathcal{B}_\alpha\} \cup \{E^c : E\in \mathcal{B}_\alpha\}$ ;
- For every ordinal limit $\lambda$, $\mathcal{B}_{\lambda}:= \cup_{\alpha<\lambda}\mathcal{B}_\alpha$.