I just proved for a homework problem that the direct product of two non-measurable sets is non-measurable. It seems to me that the finite direct product of finitely many non-measurable sets is also definitely non-measurable. But what about infinite products? I would imagine that uncountably infinitely many non-measurable sets might potentially be measurable, since uncountable products tend to do strange things. Does anyone know the conditions for when products of non-measurable sets is non-measurable? Could you point me to a proof?