Sigma Algebra and Measurable Spaces
Definitions
For a set $X \ne \emptyset$, $\mathcal{E} \subset \mathscr{P} (X)$ is called a Sigma Algebra or Sigma Field on $X$ if it satisfies the conditions below. The ordered pair $(X , \mathcal{E})$ of the set $X$ and the sigma field $\mathcal{E}$ is called a Measurable Space.
- (i): $\emptyset \in \mathcal{E}$
- (ii): $E \in \mathcal{E} \implies E^{c} \in \mathcal{E}$
- (iii): $\displaystyle \left\{ E_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{E} \implies \bigcup_{n=1}^{\infty} E_{n} \in \mathcal{E}$
- (iv): $\displaystyle \left\{ E_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{E} \implies \bigcap_{n=1}^{\infty} E_{n} \in \mathcal{E}$
Description
Given a space $X$ and a sigma field $\mathcal{E}$, $(X , \mathcal{E})$ is called a Measurable Space. If a measure $\mu$ is provided, it is called a measure space, and specifically, if the measure $\mu$ is a probability, it is referred to as a probability space.
The same concept is called Sigma Algebra in mathematics and Sigma Field in statistics.
Carathéodory’s condition: If $E \subset \mathbb{R}$ satisfies $m^{ \ast }(A) = m^{ \ast } ( A \cap E ) + m^{ \ast } ( A \cap E^{c} )$ for $A \subset \mathbb{R}$, $E$ is called a Measurable Set and is denoted as $E \in \mathcal{M}$.
A ‘Measurable Set’ simply means a set that can be measured. From the monotonicity of the outer measure, $$m^{ \ast }(A) \le m^{ \ast } ( A \cap E ) + m^{ \ast } ( A \cap E^{c} )$$ is trivial, so verifying whether a set is measurable boils down to checking if $$m^{ \ast }(A) \ge m^{ \ast } ( A \cap E ) + m^{ \ast } ( A \cap E^{c} )$$ is true.
The Sigma Algebra of Measurable Sets
With the definitions provided above, the collection $\mathcal{M}$ of measurable sets of $X = \mathbb{R}$ turns into a sigma algebra with the following properties.
$\mathcal{M}$ is a sigma algebra with the properties below.
- [1]: $$ \emptyset \in \mathcal{M} $$
- [2]: $$ E \in \mathcal{M} \implies E^{c} \in \mathcal{M} $$
- [3]: $$ \left\{ E_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{M} \implies \bigcup_{n=1}^{\infty} E_{n} \in \mathcal{M} $$
- [4]: $$ \left\{ E_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{M} \implies \bigcap_{n=1}^{\infty} E_{n} \in \mathcal{M} $$
- [5]: $$ \mathcal{N} \subset \mathcal{M} $$
- [6]: $$ \mathcal{I} \subset \mathcal{M} $$
- [7]: If we denote it as $E_{i} , E_{j} \in \mathcal{M}$, the following is true. $$ E_{i} \cap E_{j} = \emptyset , \forall i \ne j \implies m^{ \ast } \left( \bigcup_{n=1}^{\infty} E_{n} \right) = \sum_{n = 1} ^{\infty} m^{ \ast } ( E_{n}) $$
- $\mathcal{I}$ is the set of all intervals, $\mathcal{N}$ is the set of all null sets.
Especially, note that [7] is a property absolutely necessary for the ‘generalization of length’, which Lebesgue could only dream of.