  In this note, we prove a version of Tarui's Theorem in communication
complexity, namely $PH^{cc} \subseteq BP\cdot PP^{cc}$. Consequently, every
measure for $PP^{cc}$ leads to a measure for $PH^{cc}$, subsuming a result of
Linial and Shraibman that problems with high mc-rigidity lie outside the
polynomial hierarchy. By slightly changing the definition of mc-rigidity
(arbitrary instead of uniform distribution), it is then evident that the class
$M^{cc}$ of problems with low mc-rigidity equals $BP\cdot PP^{cc}$. As $BP\cdot
PP^{cc} \subseteq PSPACE^{cc}$, this rules out the possibility, that had been
left open, that even polynomial space is contained in $M^{cc}$.
