Let be a countable decomposition of and be the -algebra generated by . Are there only countably many sets in ?
The answer is no. We can show this by showing that the natural numbers is a strict subset of . Every natural number can be written as where and only a finite number of (because if an infinite number of the sum is ).
Note that since is a decomposition we can write every set in as a countable (because this is a -algebra) union of a subset of . This means we can encode every set in as where . First, since is countable there is a bijection between the natural numbers and , however, is not countable since we can have a countable number of .