$\mathsf{P}^\mathsf{BPP}$ vs $\mathsf{BPP}$ (Are they known to be equal) Is it known if $\mathsf{P}^\mathsf{BPP}= \mathsf{BPP}$ ?


It's clear that $\mathsf{BPP} \subseteq \mathsf{P}^\mathsf{BPP}$. Now, since $\mathsf{BPP}$ is closed under complementation, union, and intersection, it seems to provide evidence that $\mathsf{BPP} = \mathsf{P}^\mathsf{BPP}$. However, I can't prove this.
Yes, since BPP is low, but this is not a research level question.