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Why closedness of complement of randomized classes imply containment of complement of contained classes?

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Problem Detail: 

Suppose if class $\mathcal C$ is in $PP$ or $BPP$ does it mean complement also belongs to $PP$ or $BPP$ respectively? Does it immediately follow from $PP=coPP$?

Asked By : Turbo
Answered By : Ariel

Yes, you should follow the definitions, if $L\in \mathcal{C}\subseteq PP$, then $\overline{L}\in PP$. It immediately follows that $co-\mathcal{C}=\{L | \overline{L}\in \mathcal{C}\}\subseteq PP$.

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