Probability sigma field

In summary, a probability sigma field, or sigma algebra, is a collection of subsets of a sample space that is used to define the notion of probability for events. It must contain the empty set and the entire sample space, be closed under complementation, and closed under countable unions. It is specifically used in probability theory and is important because it allows us to define the probability of events. In practical applications, it is used to determine the probability of events and make informed decisions based on that probability.
  • #1
rukawakaede
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Homework Statement


Let [tex](\Omega,\mathcal{F},P)[/tex] be a probability space. Let [tex]\mathcal{N}\subseteq\mathcal{P}(\Omega)[/tex] where [tex]\mathcal{N}:=\{N\subseteq\Omega:\exists M\in\mathcal{F},P(M)=0[/tex] and [tex]N\subseteq M\}[/tex].

(a) Want to show that [tex]\mathcal{F}'=\{F\cup N:F\in\mathcal{F}, N\in\mathcal{N}\}[/tex] defines a [tex]\sigma[/tex]-field. **only closed under complement part.

(b) For [tex]F,F'\in\mathcal{F}[/tex] and [tex]N,N'\in\mathcal{N}[/tex] with [tex]F\cup N=F'\cup N'[/tex] we have [tex]P(F)=P(F')[/tex].

* I need some directions on how to approaching these.

Homework Equations


All information given at (1.)

The Attempt at a Solution


(a) this part I only need assistance on the complement closure part. My approach is first show the empty set is in [tex]\mathcal{N}[/tex] then show that [tex]F'^c=(F\cup N)^c=F^c\cap N^c\in\mathcal{F}'[/tex]. Initially I thought I would be just fine to use Continuity from below to show that [tex]N^c=\Omega[/tex] but this is completely false, since [tex]N[/tex] can be non empty and it is in [tex]\mathcal{N}[/tex] not in [tex]\mathcal{F}[/tex]. Continuity lemma does not apply.

(b) for this part I made a fault direct attempt. I intended to show [tex]F\cup N=F'\cup N'[/tex] implies [tex]P(F\cup N)=P(F'\cup N')[/tex] but this is totally wrong too since [tex]P[/tex] is not define for [tex]N\in\mathcal{N}[/tex].

Thanks in advanced.
 
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  • #2


For part (a), you can use the fact that \mathcal{F} is a \sigma-field to show that \mathcal{F}' is closed under complements. First, note that for any F\in\mathcal{F} and N\in\mathcal{N}, we have F\cup N\subseteq F^c. This is because if x\in F\cup N, then either x\in F or x\in N. But since N\subseteq F^c, we have x\in F^c. Therefore, F\cup N is a subset of F^c.

Next, we need to show that F^c\in\mathcal{F}'. Since F^c is a complement of F, we know that F^c\in\mathcal{F}. Now, we need to find an N\in\mathcal{N} such that F^c=N\cup F. Consider the set N=F^c\setminus F. This set is in \mathcal{N} because it is a subset of F^c and P(F^c)=0. Also, note that N\cup F=F^c, which means that F^c\in\mathcal{F}'.

For part (b), you can use the fact that if two sets have the same measure, then their union also has the same measure. So, if F\cup N=F'\cup N', then we have P(F\cup N)=P(F'\cup N'). Now, note that F\subseteq F\cup N, which means that P(F)\leq P(F\cup N). Similarly, we have P(F')\leq P(F'\cup N'). But since P(F\cup N)=P(F'\cup N'), we have P(F)=P(F').
 

1. What is a probability sigma field?

A probability sigma field, also known as a sigma algebra, is a collection of subsets of a sample space that satisfies certain properties. It is a fundamental concept in probability theory and is used to define the notion of probability for events.

2. What are the properties of a probability sigma field?

A probability sigma field must contain the empty set and the entire sample space, be closed under complementation, and closed under countable unions. This means that any event that can be defined within the sample space must be included in the sigma field, and the sigma field must also include the complement of any event in the sigma field, as well as the union of countably many events in the sigma field.

3. How is a probability sigma field different from a sigma field?

A probability sigma field is a type of sigma field that is specifically used in probability theory. While a sigma field can be defined for any set, a probability sigma field is only defined for a sample space. Additionally, the properties of a probability sigma field are tailored to the concept of probability, while a general sigma field may have different properties.

4. Why is a probability sigma field important in probability theory?

A probability sigma field is important because it allows us to define the probability of events within a sample space. Without a sigma field, we would not have a way to determine the probability of events or analyze the likelihood of certain outcomes in a probabilistic setting.

5. How is a probability sigma field used in practical applications?

In practical applications, a probability sigma field is used to determine the probability of certain events occurring in a given scenario. It is also used in decision making, risk analysis, and statistical modeling to assess the likelihood of certain outcomes and make informed decisions based on that probability.

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