Sigma-algebra

Definition

Let X be some set, and let 2^X represent its power set. Then a subset $\Sigma \subseteq 2^{X}$ is called a σ-algebra if it satisfies the following three properties:.

1. X is in ${\mathcal {S}}$, and X is considered to be the universal set in the following context.
2. Σ is closed under complementation: If A is in Σ, then so is its complement, X \ A.
3. ${\mathcal {S}}$ is closed under countable unions: If A₁, A₂, A₃, ... are in ${\mathcal {S}}$, then so is

$A=A_{1}\cup A_{2}\cup A_{3}\cup \ldots =\bigcup _{i=1}^{\infty }A_{i}$

Lemmas

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).

It also follows that the empty set $\emptyset$ is in ${\mathcal {S}}$, since by

• (1) X is in ${\mathcal {S}}$
• (2) asserts that its complement, the empty set, is also in ${\mathcal {S}}$. Moreover, since $\{X,\emptyset \}$ satisfies condition
• (3) as well, it follows that $\{X,\emptyset \}$ is the smallest possible σ-algebra on X.
• The largest possible σ-algebra is the power set on X, which contains is $2^{n}$ elements, if X is finite and contains n elementss.

Elements of the $\sigma$-algebra are called measurable sets. An ordered pair $(X,{\mathcal {S}})$, where X is a set and ${\mathcal {S}}$ is a $\sigma$-algebra over X, is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a $\sigma$-algebra to [0, ∞].

A $\sigma$-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see Wikipedia:Sigma algebra).

Learning Task

• Explore the measurement problem and explain why a $\sigma$-algebra is necessary for the definition of probability distributions!
• Explain why a discrete probability distribution can use the