Let $A,B$ be two Boolean algebra with measures $m,p$ thereon, respectively such that the measure algebra $(A,m)$ is isomorphic to the measure algebra $(B,p)$. Suppose that we have two isomorphic infinite (atomless) subalgebras $C\subset A$ and $D\subset B$.
Can we have isomorphic, say, measure subalgebras, i.e. $(C,m|_C)$ isomorphic to $(D,p|_D)$.
Any idea or references are welcome.
Edit: After seeing Gogi Pantsulaia's example I found that I forgot to mention a word in my assumption.
Let $A=B=B([0,1])$ be Borel $\sigma$ algebra of $[0,1]$ with measure $m=p=l_1$, where $l_1$ denotes linear Lebesgue measure. Then $(A,m)=(B,p)$, hence these measure algebras are isomorphic. Let consider two algebras $C=\{[0,1], K, [01]\setminus K, \emptyset \}$, where $K$ is Cantor set, and
$C=\{[0,1], [0,1/2],]1/2,1], \emptyset\}$. It is obvious that $C$ and $D$ are Borel isomorphic. Now notice that measure algebras $(C,m|C)$ and $(D,p|D)$ are not isomorphic.