A sigma field, also known as a sigma algebra, is a collection of subsets of a given set that satisfies certain properties. It is an important concept in measure theory and probability theory.
To form a sigma field from a countable infinite set, we must take the power set of the given set, which is the set of all possible subsets. Then, we must take the sigma field generated by this power set, which is the smallest sigma field that contains all subsets of the given set.
A sigma field must satisfy three properties: it must contain the empty set and the entire set, it must be closed under complement (if a set is in the sigma field, its complement must also be in the sigma field), and it must be closed under countable unions (if a countable number of sets are in the sigma field, their union must also be in the sigma field).
Forming a sigma field from a countable infinite set is important because it allows us to define measures and probabilities on this set. It also allows us to apply important theorems and results from measure theory and probability theory, which are crucial in many areas of science and mathematics.
Yes, a sigma field can be formed from an uncountable set. The process is the same as forming a sigma field from a countable infinite set, but instead of taking the power set, we take the sigma field generated by the set of all subsets. This will still satisfy the properties of a sigma field and allow for the definition of measures and probabilities on the uncountable set.