Can a sigma algebra be countable?

Definition of sigma algebra [3]: a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections. Assume a countably infinite sigma algebra does exist.

Is Borel sigma algebra Countably generated?

The σ-algebra of Borel subsets of M will be denoted by B. A measurable space (X, E) is said to be countably generated if E = σ(S) for some countable subset S of E and is said to be separable if {x}∈E for each x ∈ X. In particular, a standard Borel space is both countably generated and separable.

How do you construct the smallest sigma algebra?

To obtain the smallest σ-algebra containing it, all you need to do is add the missing sets that make it a σ-algebra (instead of just being a set). What this means is that you want to add all sets so that the resulting set is closed with respect to taking complements and union.

Why sigma-algebra is measurable?

The reason is that the σ-algebra of subsets determines the σ-algebra of measurable real functions, and vice versa. Thus A ⊆ F is a measurable subset if and only if the indicator function 1A is a measurable real function with respect to F.

What generates the Borel sigma algebra?

The Borel σ-algebra b is generated by intervals of the form (−∞,a] where a ∈ Q is a rational number. σ(P) ⊆ b. This gives the chain of containments b = σ(O0) ⊆ σ(P) ⊆ b and so σ(P) = b proving the theorem.

What is Borel set example?

Here are some very simple examples. The set of all rational numbers in [0,1] is a Borel subset of [0,1]. More generally, any countable subset of [0,1] is a Borel subset of [0,1]. The set of all irrational numbers in [0,1] is a Borel subset of [0,1].

What is a generated sigma algebra?

The generated σ-algebra or generated σ-field refers to. The smallest σ-algebra that contains a given family of sets, see Generated σ-algebra (by sets) The smallest σ-algebra that makes a function measurable or a random variable, see Sigma-algebra#σ-algebra generated by a function.

Are sigma algebra closed under countable intersections?

A σ-algebra is a non-empty set of sets that is closed under countable unions, countable intersections, and complements. In other words, if An,n∈N reside in a σ-algebra A then we also have ∪n∈NAn∈A, ∩n∈NAn∈A and Acn∈A.

Are sigma-algebra closed under countable intersections?

What are countable sets examples?

Examples of countable sets include the integers, algebraic numbers, and rational numbers. Georg Cantor showed that the number of real numbers is rigorously larger than a countably infinite set, and the postulate that this number, the so-called “continuum,” is equal to aleph-1 is called the continuum hypothesis.