Sigma-algebra

Definition
Let X be some set, and let 2X 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 $$\{X,\empty\}$$ satisfies condition
 * (3) as well, it follows that $$\{X,\empty\}$$ 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 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