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 .