Let $(\Omega_1,\mathcal {F_1})$ and $(\Omega_2,\mathcal {F_2})$ be two measurable spaces.we call $\mathcal{F_1}$ and $\mathcal {F_2}$ are isomorphic if there exist a measurable map $f: (\Omega_1,\mathcal {F_1}) \to (\Omega_2,\mathcal {F_2})$ such that the natural map $f_{*}: \mathcal {F_2} \to \mathcal {F_1}$ is bijective and preserves arbitrary union and complements.
Let $(\Omega,\mathcal {F})$ $( \vert \mathcal {F} \vert < \infty)$ and $(\{0,1\}^k,\mathcal {P}(\{0,1\}^k))$ be measurable spaces. Does there always exist a $k$ such that $\mathcal {F}$ is isomorphic to $\mathcal {P}(\{0,1\}^k)$?
I know that cardinality of $\mathcal {F}$ is $2^n$ for some $n$.Any ideas to proceed?