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What is sigma-field in probability?
A sigma-field refers to the collection of subsets of a sample space that we should use in order to establish a mathematically formal definition of probability. The sets in the sigma-field constitute the events from our sample space.
What is a generated sigma-algebra?
The generated σ-algebra or generated σ-field refers to. The smallest σ-algebra that contains a given family of sets, see Generated σ-algebra (by sets) The smallest σ-algebra that makes a function measurable or a random variable, see Sigma-algebra#σ-algebra generated by a function.
Why do we need sigma algebras in probability?
Sigma algebra is necessary in order for us to be able to consider subsets of the real numbers of actual events. In other words, the sets need to be well defined, under the conditions of countable unions and countable intersections, for it to have probabilities assigned to it.
How do you prove smallest sigma-algebra?
The term “smallest” here means that any sigma-algebra containing the sets of B would have to contain all the sets of σ(B) as well. ∩G = {A ⊂ X| A ∈ F for every F ∈ G} consists of all sets A which belong to each sigma-algebra F of G.
Is every sigma-algebra Borel?
The reason for this distinction is that the Borel sets are the σ-algebra generated by open sets (of a topological space), whereas Mackey’s definition refers to a set equipped with an arbitrary σ-algebra. There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.
What’s the use of sigma-algebra?
Which is an example of a sigma-algebra induced by a random variable?
For the reason that we are dealing with preimages of the random variable X, we call Σ the sigma-algebra induced by the random variable X. Here is an extreme example: consider a constant random variable X, that is, X(ω) ≡ α. Then X − 1(B), B ∈ B(R) equals either Ω or ∅ depending on whether α ∈ B.
Which is the smallest algebra that makes X a random variable?
Σ is in fact the smallest sigma-algebra that makes X a random variable as all other sigma-algebras of that kind would at the very least include Σ. For the reason that we are dealing with preimages of the random variable X, we call Σ the sigma-algebra induced by the random variable X.
Is the sigma algebra trivial in cross validated?
The sigma-algebra thus generated is trivial and as such, it is definitely included in A. Hope this helps. Thanks for contributing an answer to Cross Validated!
How to show that σ is a sigma algebra?
Using the properties of preimages, it is not too difficult to show that Σ is a sigma-algebra. It also follows immediately that Σ ⊂ A, hence Σ is a sub-sigma-algebra. Further, by the definitions it is easy to see that the mapping X: (Ω, Σ) → (R, B(R)) is measurable.