If is a -algebra for , where is non-empty, then their intersection is also a -algebra. Indeed it follows directly from properties , and .
For any , it then follows from the proposition above that their exists a unique smallest -algebra that contains , denoted , i.e. for any -algebra that contains , . This can be proved explicity via construction.
Let denote the set of -algebra’s that contain . Clearly is non-empty as . Then it must hold that the intersection is also a -algebra. Since for all , it follows that this intersection is itself contains . Thus it follows that exists and is equal to . Such is called the -algebra generated by .
Lemma
If , then .
This follows since is a -algebra containing -therefore it must contain .