來練習一下定理和定義的默寫好了

F is called a field if:
1. omega belong to F
2. A belong to F implice complete of A belong to F
3. exist a finite sequence belong to F such that their union or intersection belong to F

F is called a sigma-field if:
1. omega belong  to F
2. A belong to F implice the complete of A belong to F
3. exist countable sequence which belong to F such that their unoin or intersection belong to F

[iframe]http://www.youtube.com/watch?v=Irddbq3o-Zc&feature=related[/iframe]

If F is a field and closed under limit of increasing sequence
then F is a sigma-field

Let C be the collection of subets of omega
sigma(C) is the minimal sigma-field which containing C
sigma(C) := intersection{F: F is a sigma-filed, F containing C}

a sigma-field on omega intersect A is a sigma-field on A
sigma(C) intersect A = sigma(C intersect A)
       on omega                  on A

Definition:
Let F be a field on omega
Let mju is a set function from F to R
1. mju is c.a. :
    for each An belong to F s.t. unions of An belong to F, mju(unoins of An)=Sum(mju(An))
2. If mju is c.a. and mju is nonnegative, then mju is a measure
3. If mju is a measure and mju(omega)=1 then mju is a probability measure
4. (omega, sigma-field, mju is a measure) is called a measure space
5. Let omega=R, Borel sigma-field is the smallest sigma-field on R which containing all right-semiclosed intervels