Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets.

The Cohen algebra has a combinatorial : it is the unique atomless complete Boolean algebra with a countable dense subset, (i.e it is a completion of a countable dense Boolean subalgebra). This case is true when $X=2^\omega$.

(1) My first question is: does this characterization true when $X=2^\kappa$ for an infinite $\kappa$.

(2) Is there any characterization (unique like above one) for random algebra (= Bor($X$) modulo the ideal of Lebesgue measure zero set) even when when $X=2^\omega$.

Thanks in advance