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Sigma Algebra and Measurable Spaces 📂Measure Theory

Sigma Algebra and Measurable Spaces

Definitions

For a set X≠∅X \ne \emptyset, E⊂P(X)\mathcal{E} \subset \mathscr{P} (X) is called a Sigma Algebra or Sigma Field on XX if it satisfies the conditions below. The ordered pair (X,E)(X , \mathcal{E}) of the set XX and the sigma field E\mathcal{E} is called a Measurable Space.

  • (i): ∅∈E\emptyset \in \mathcal{E}
  • (ii): E∈E  ⟹  Ec∈EE \in \mathcal{E} \implies E^{c} \in \mathcal{E}
  • (iii): {En}n∈N⊂E  ⟹  ⋃n=1∞En∈E\displaystyle \left\{ E_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{E} \implies \bigcup_{n=1}^{\infty} E_{n} \in \mathcal{E}
  • (iv): {En}n∈N⊂E  ⟹  ⋂n=1∞En∈E\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 XX and a sigma field E\mathcal{E}, (X,E)(X , \mathcal{E}) is called a Measurable Space. If a measure μ\mu is provided, it is called a measure space, and specifically, if the measure μ\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⊂RE \subset \mathbb{R} satisfies m∗(A)=m∗(A∩E)+m∗(A∩Ec)m^{ \ast }(A) = m^{ \ast } ( A \cap E ) + m^{ \ast } ( A \cap E^{c} ) for A⊂RA \subset \mathbb{R}, EE is called a Measurable Set and is denoted as E∈ME \in \mathcal{M}.

A ‘Measurable Set’ simply means a set that can be measured. From the monotonicity of the outer measure, m∗(A)≤m∗(A∩E)+m∗(A∩Ec)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∗(A)≥m∗(A∩E)+m∗(A∩Ec)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 M\mathcal{M} of measurable sets of X=RX = \mathbb{R} turns into a sigma algebra with the following properties.

M\mathcal{M} is a sigma algebra with the properties below.

  • [1]: ∅∈M \emptyset \in \mathcal{M}
  • [2]: E∈M  ⟹  Ec∈M E \in \mathcal{M} \implies E^{c} \in \mathcal{M}
  • [3]: {En}n∈N⊂M  ⟹  ⋃n=1∞En∈M \left\{ E_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{M} \implies \bigcup_{n=1}^{\infty} E_{n} \in \mathcal{M}
  • [4]: {En}n∈N⊂M  ⟹  ⋂n=1∞En∈M \left\{ E_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{M} \implies \bigcap_{n=1}^{\infty} E_{n} \in \mathcal{M}
  • [5]: N⊂M \mathcal{N} \subset \mathcal{M}
  • [6]: I⊂M \mathcal{I} \subset \mathcal{M}
  • [7]: If we denote it as Ei,Ej∈ME_{i} , E_{j} \in \mathcal{M}, the following is true. Ei∩Ej=∅,∀i≠j  ⟹  m∗(⋃n=1∞En)=∑n=1∞m∗(En) 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})
  • I\mathcal{I} is the set of all intervals, N\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.