
> 
> 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



Cohen algebra (generalization)
Regarding question $1$, it seems that you want to know whether
you've got the unique complete c.c.c. Boolean algebra with density
$\kappa$. The answer is no.


On the one hand, the forcing notion $\text{Add}(\omega,\omega\_1)$
to add $\omega\_1$ many Cohen reals is c.c.c. and has density
$\omega\_1$. This is another way to describe your algebra in the case $\kappa=\omega\_1$.


But meanwhile, forcing with a Suslin tree is also c.c.c. and has
density $\omega\_1$. But these two complete Boolean algebras are not
isomorphic, since forcing with the Suslin tree adds no reals and in fact is
$\leq\omega$-distributive, whereas clearly
$\text{Add}(\omega,\omega\_1)$ adds reals.


Of course, there may not be a Suslin tree, but consider the
forcing to add a single Cohen real (which creates a Suslin tree),
followed by the forcing to force with that Suslin tree. This is
the iteration of two c.c.c. forcing notions, and is hence c.c.c.,
but under CH has density $\omega\_1$. But this forcing is not
isomorphic to $\text{Add}(\omega,\omega\_1)$, since the latter
forcing is absolutely c.c.c., but the former is not, as the branch
through the Suslin tree creates an uncountable antichain in the
extension.


One may omit the CH assumption by taking the case
$\kappa=\mathfrak{c}$. That is, compare the forcing to add
continuum many Cohen reals, versus the forcing to add this many
Cohen reals, and afterwards also to force with the Suslin tree
created by the first one. Both of these are c.c.c. and have
density $\mathfrak{c}$, but they are not isomorphic since the
former remains c.c.c. in the forcing extension and the latter does
not.


