Stochastic Meaning Theory
Continuity of Meaning
12th for KARCEVSKIJ Sergej
1
Set X
Family of subsets of X M
When M satisfies the next <1>(i)(ii)(iii), M is called σ-field.
<1>
(i) X, Ø ∈M
(ii) a∈M ⇒ X╲A∈M
(iii) An∈M (n=1, 2, …) ⇒∪∞n=1 An∈M
( X, M ) is called measurable space.
Function over M μ
When μ satisfies the next <2>(i)(ii)(iii), μ is called measure over measurable space ( X, M ).
(i) μ (A)∈[0,∞]
(ii) μ (0) = 0
(iii) An∈M , An ∩Am = 0 (n≠m)
μ (∪∞n=1 An) = Σ∞n=1 μ (A)
( X, M, μ ) is called measure space.
When measure space satisfies the next <3>(i), it is called complete measure space.
(i) A∈M, μ (A) = 0 ⇒ B⊂A, μ (B) = 0
<2>(iii) is called complete additive or σ additive.
2
Measure space that is all the measure is 1 is called probability space.
Measure that all the measure is 1 is called probability measure.
3
Set Ω that is called whole possibility
Element of Ω ω that is called sample point
σ-field F
Element of F A that is called event
Function over F P
Measure for A∈F P (A ) that is called probability
4
Probability space ( Ω, F, P )
valued function over Ω X
When X is F- measurable, it is called random variable.
When value of measurable space (S, M) is not but S, variable is called S valued random variable.
Family of subsets of Ω { An }∞n=1
When { An }∞n=1 satisfies the next <1>(i)(ii), it is called countable decomposition of Ω.
(i) An ∩ Am = Ø ( n ≠ m )
(ii) ∪∞n=1 An = Ω
5
Almost countable set S that has σ-field
Separable space ( Ω, F )
Sequence of S valued random variable {Xn}∞n=1
Sub-σ-field of F Fn : = σ ( Xk ; 0 ≤k ≤1)
x, y ∈S
0 ≤p (x, y) ≤1
WhenΣ y ∈S p(x, y) = 1, x∈S is satisfied, p is called transition probability.
Family of probability measure {Pz}z∈S
When {Xn}∞n=1 and {Pz}z∈S is satisfies the next <2>(i)(ii) for bounded function over S, they are called Markov chain that has transition probability p.
<2>
(i) Pz ( X0 = x ) = 1
(ii) Ex ( f (Xn+1 ) | Fn ) =Σ y ∈S p (Xn, y) f(y) a. s. Px
<2>(ii) is called Markov
Tokyo June 22, 2008
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