Is Axiom 2 of Sigma Algebras Equivalent to X Being in the Sigma Field?

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In summary, a sigma algebra (or sigma field) is a subset \Sigma of the powerset of a set X that satisfies the following axioms:1) E\in \Sigma \Rightarrow E^c \in \Sigma2) E_i \in \Sigma \ \ \forall i \in I \Rightarrow \bigcup_{i\in I}E_i \in \Sigma (where the index set I is countable)It is noted that axiom 2 can be simplified to "2') X\in \Sigma" since any element of \Sigma and its complement will result in the union being X itself. However, axiom 2 also implies that X is in \Sigma by taking an element of \Sigma
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quasar987
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Wiki says that a sigma algebra (or sigma field) is a subset [itex]\Sigma[/itex] of the powerset of some set X satisfying the following axioms

1) [tex]E\in \Sigma \Rightarrow E^c \in \Sigma[/tex]

2) [tex]E_i \in \Sigma \ \ \forall i \in I \Rightarrow \bigcup_{i\in I}E_i \in \Sigma[/tex]

(where the index set I is countable)

Am I missing something or is axiom 2 equivalent to the much less complicated "2') [itex]X\in \Sigma[/itex]"? Cause for any element of [itex]\Sigma[/itex], since its complement is in [itex]\Sigma[/itex] also, the union of both is X itself. So 2) is satified as soon as 2') is. Conversely, 2) implies that X is in [itex]\Sigma[/itex] simply by taking an element of [itex]\Sigma[/itex] and its complement in the union.
 
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(2) applies to any sequence of elements of sigma. That includes sequences that don't contain a pair of complementary sets.
 
  • #3
oh, right!
 

Related to Is Axiom 2 of Sigma Algebras Equivalent to X Being in the Sigma Field?

What are sigma-algebras and why are they important?

Sigma-algebras are mathematical structures used in probability theory and measure theory. They are important because they provide a way to define and measure sets, which is essential in understanding and solving problems in these fields.

What are the axioms of sigma-algebras?

The axioms of sigma-algebras are a set of mathematical rules that must be satisfied in order for a collection of sets to be considered a sigma-algebra. These axioms include closure under countable unions and intersections, closure under complements, and containing the underlying space.

What is the difference between a sigma-algebra and a regular algebra?

A sigma-algebra is a more general and powerful concept than a regular algebra. While a regular algebra only requires closure under finite unions and intersections, a sigma-algebra requires closure under countable unions and intersections. This allows for more complex and abstract sets to be defined and measured.

How are sigma-algebras used in probability theory?

In probability theory, sigma-algebras are used to define measurable events and calculate their probabilities. They also allow for the construction of probability measures, which are essential in understanding the likelihood of certain outcomes in a given system.

Can the axioms of sigma-algebras be violated?

Yes, the axioms of sigma-algebras must always be satisfied in order for a collection of sets to be considered a sigma-algebra. If any of the axioms are violated, the structure is no longer a sigma-algebra and cannot be used in probability or measure theory.

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