Sigma Algebras on [0,1]: Understanding Measures and Sub-Intervals

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In summary, the conversation discusses the formation of a set A from a semi-algebra J, where J is the collection of all intervals in [0,1]. It is proven that A is an algebra, but it is questioned why it is not a σ-algebra. The conversation then leads to discussing if singletons and rationals are in A, with the conclusion that all {r} in A implies that Q∩[0,1] is also in A. However, there is still uncertainty about whether or not Q is in A.
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Old Monk
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This topic came up while studying measures on sub-intervals of [0,1]. The collection of all intervals in [0,1] is a semi-algebra, say J. Now from finite disjoint union of members of J let's say we form a set A.

I was able to prove that A is an algebra, since for any C,D ε A, C[itex]\cap[/itex]D and C[itex]^{c}[/itex] belong to A.

I'm not able to understand why A isn't a σ-algebra. Can anyone please outline a proof or give me a counter-argument.
 
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  • #2
Are singletons in A?? Is [itex]\mathbb{Q}[/itex] in A?
 
  • #3
Yeah singletons are in A. Since singletons of the form (a,a) exist in J, the single union of such elements should exist in A. That I assume implies, all the rationals in [0,1] are also in A.
 
  • #4
Old Monk said:
Yeah singletons are in A. Since singletons of the form (a,a) exist in J, the single union of such elements should exist in A.

Isn't [itex](a,a)=\emptyset[/itex]?? Singletons would rather come from [itex][a,a]=\{a\}[/itex]. But ok, let's assume that the singletons are in A.


That I assume implies, all the rationals in [0,1] are also in A.

Why??
 
  • #5
Yeah sorry, that's [a,a].

We have [0,r)[itex]\cup[/itex](r,1] in A, if r is any rational in [0,1]. Therefore the element {r} ε A. So I think we could generalize this to all r ε Q[itex]\cap[/itex][0,1].
 
  • #6
Old Monk said:
Yeah sorry, that's [a,a].

We have [0,r)[itex]\cup[/itex](r,1] in A, if r is any rational in [0,1]. Therefore the element {r} ε A. So I think we could generalize this to all r ε Q[itex]\cap[/itex][0,1].

Yeah, I agree that every {r} is in A. But why does that imply that [itex]\mathbb{Q}\in A[/itex]. Is it even true??
 

Related to Sigma Algebras on [0,1]: Understanding Measures and Sub-Intervals

1. What is a sigma algebra on [0,1]?

A sigma algebra on [0,1] is a collection of subsets of the interval [0,1] that satisfies certain properties, such as being closed under countable unions and complements. It is a fundamental concept in measure theory and plays a crucial role in defining measures and integrating functions.

2. What are some examples of sigma algebras on [0,1]?

Some examples of sigma algebras on [0,1] include the Borel sigma algebra, which is generated by the open intervals on [0,1], and the Lebesgue sigma algebra, which is generated by the Lebesgue measurable sets on [0,1]. Other examples include the power set of [0,1] and the trivial sigma algebra, which consists of only the empty set and the entire set [0,1].

3. Why are sigma algebras important in probability theory?

Sigma algebras are important in probability theory because they provide a way to define and measure the likelihood of events. In probability theory, a sigma algebra represents the set of all events for which probabilities can be assigned. This allows for the precise calculation of probabilities and the development of important concepts such as conditional probability and independence.

4. How are sigma algebras related to measurable functions?

Sigma algebras are closely related to measurable functions because they provide the necessary structure for defining such functions. In particular, a measurable function is one that preserves the structure of the sigma algebra, meaning that the preimage of any measurable set is also measurable. This allows for the integration of functions with respect to a given measure on the sigma algebra.

5. Can a sigma algebra on [0,1] be infinite?

Yes, a sigma algebra on [0,1] can be infinite. In fact, most commonly used sigma algebras on [0,1] are infinite, such as the Borel sigma algebra and the Lebesgue sigma algebra. This is because there are infinitely many subsets of [0,1] and the sigma algebra must contain all of these subsets in order to be closed under countable unions and complements.

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