Sigma Algebra

Let be a set and be a family of sets over . Then is a sigma-algebra (-algebra) of subsets of if and only if it satisfies the following properties:

2. is closed under complement:
   • If , then
3. is closed under countable unions:
   • If is a countable collection of elements of and , then

In the context of a probability space, we say that and .

If , then one possible -algebra on is .

However, is not a -algebra because it does not contain the complement of or .

There are two extreme examples of a sigma algebra:

• The collection
• The power set of ,

Any sigma algebra lies between these two extremes. That is, .

Credit: some stack exchange answer.
Focusing specifically on the application to probability:

The first property states that we always know the probability of something happening (1) and nothing happening (0)

For the second property, suppose we have a coin flip with . If we have a "-algebra" , we would know the probability of nothing happening, something happening, and the probability of flipping a heads. But according to the given -algebra, we don't know the probability of tails (the complementary event of heads). This is quite moronic. If we know the probability of something happening, then you automatically know the probability of it not happening. Hence, -algebras must be closed under complementation.

For the third property, suppose we roll a die, so . If we give a "-algebra" , we know the chance of rolling a 1, and we know the chance of rolling a 2. But according to the given -algebra, we don't know the probability of rolling a 1 or 2. This is once again quite moronic. If we know the probability of two events happening, we know the probability of either of them happening. If the events happen not to be disjoint, it's a simple matter of using the additive principle for the union of sets.

This may seem like the event space. In fact, the event space is a -algebra.

An atom of a -algebra is a set such that the only subsets of are also in are the empty set and itself.

In other words, an atom of a -algebra is any set in the -algebra which have cardinality 1.

Consider the set , and (the power set of ), then the atoms would be .

The collection consisting of all unions of the partition sets forms a -algebra.

Consider the set , and partitions , , and .

Then the sigma algebra formed by these partitions is:

Let be a set and be a non-empty family of sets of . Then the smallest -algebra containing all the sets of is called the -algebra generated by the collection and we denote is as .

In other words, the sigma algebra generated by contains all the sets of , as well as any other sets required to make it a valid -algebra, and nothing more.

Given , and , then .

The first two sets satisfy are requirements of a -algebra; the empty set as well as the entire set . Note that while is in , it doesn't have to me.

The next set () is the first set of , and the set after is its complement ().

The next set () is the second set of , and the set after is its complement ().

The final set () ensures that the -algebra is closed under countable unions. That is, any "combination" of unions must be included. If we do , we get . Any other union combination already exists in the sigma algebra.

Finally, this is the smallest -algebra because it does not contain any elements which are not in and are not necessary to satisfy the properties of a -algebra.

If is a non-empty collection of -algebras of , then the intersection of all the elements of () is also a -algebra of .

If you take the intersection of all -algebras, you're left with the smallest one (all -algebras need the bare minimum)

A collection of subsets of is a π-system if is closed under finite intersections (if , then ).

A collection of subsets of is called a λ-system if:

1. contains the empty set ()
2. is closed under complements:
   • If then
3. is closed under countable disjoint unions
   • If and for all , then

If where is a -system, then as well

A family which is both a -system and a -system is also a -algebra

If is a -system and is a -system of subsets of , then:

That is, the sigma algebra generated by is contained in .