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Definitions IV (Continuous)

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A nonempty collection of subsets of a set Ω that includes Ω itself and is closed under the complement and countable unions (and intersections) of its members.
Sigma-algebra generated by E
The smallest sigma-algebra which contains every set of a collection E of subsets of Ω.
Measurable space
A pair (Ω,Σ) of a non-empty set Ω and a sigma-algebra Σ of subsets of Ω.
Measurable set
A set contained in the sigma-algebra Σ of a measurable space (Ω,Σ).
Probability measure
A sigma-additive function P : Σ → [0,1] on a measurable space (Ω,Σ) such that P(Ω) equals 1.
Probability triple (Continuous probability space)
A triple (Ω,Σ,P) of a non-empty set Ω, a sigma-algebra Σ of subsets of Ω and a probability measure on (Ω,Σ).
Measurable function
A map between measurable spaces (Ω,Σ) and (Ω',Σ') such that the inverse image of all sets in Σ' is contained in Σ.
Pretty much everything (combination of measurable functions)
What can you do to measurable functions without leaving the domain of measurable functions?
Continuous random variable
A measurable function defined on the sample space Ω of a probability triple (Ω,Σ,P).
Discrete sigma-algebra
The power set of a set Ω viewed as algebra of subsets.
Borel algebra (in general)
Sigma-algebra generated by the open (or closed) sets of a topological space.
Borel algebra (in R^n)
Sigma-algebra generated by the half-open intervals in R^n.
Trivial sigma-algebra
The algebra of subsets of a set Ω consisting only of the empty set and the set Ω.
Sigma-algebra generated by singletons
The collection of subsets of a set Ω which are countable or whose complements are countable.
Sigma-algebra generated by a function
For a function from a measurable space (Ω,Σ) to a set Ω': the family of subsets of Ω' whose inverse images are contained in Σ.