It can be shown that any infinite sigma algebra has cardinality greater than or equal to that of the continuum.
You probably need to use the axiom of choice. Let X be a set and an infinite sigma algebra over X. Then define the equivalence relation ~ as x~y if and only if . Then each equivalence class E(x) is measurable, as it is the countable intersection of sets in the sigma-algebra containing x.
Let I be the set of representatives from each equivalence class (Axiom of Choice here). Then is a disjoint union. Hence if we suppose that is countable then I must either be finite or countable. Thus we have an injection from to as we can pick an element of by taking (as they are equivalence classes, you do have a guarantee of getting distinct sets).
This is impossible as if I is countable or finite.
here is another proof which takes a different approach.