Iterating the Big--Pieces operator and larger sets

We show that if an Ahlfors-David regular set $E$ of dimension $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by $BP(BP(LG))$), then $E\subset \tilde{E}$ where $\tilde{E}$ is Ahlfors-David regular of dimension $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by $BP(LG)$. Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors-David regular sets $\mathcal{F}$ replacing $LG$. A simple corollary is the stability of the BP operator after 2 iterations. This was previously only known in the Euclidean setting for the case $\mathcal{F}= LG$ with substantially more complicated proofs.