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
Tokyo June 22, 2008
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