Let mju be a f.a. set ftn.
1. If mju is continuous from below at each A, then mju is c.a.
   ie. An belong to F s.t. An increasing to A and mju(An) increasing to mju(A)
2. If mju is continuous form above at empty set, then mju is c.a.
   ie. An belong to F s.t. An decreasing to empty set, and mju(An) decreasing to 0

G is the collection of all limits of increasing sequence of set in F0
G := {A in omega: exist An in F0 s.t. An increasing to A}

mju_star(A) := inf {mju(g): g in G, g containing A}

H := {h in P(omega): mju_star(h)+mju_star(complete of h) = 1}

Thus
1. G in H
2. H is a sigma-field
3. mju_star is a probability measure on H

mju is complete in F if:
for each A in F, mju(A) implice any subsets of A belong to F

the completion of (omega, sigma-field, mju) is (omega, sigma-field_mju, mju_bar)
which sigma-field_mju := {A union N : A belong F, exist M belong to F s.t. mju(M)=0 and N in M}
mju_bar(A union N) := mju(A) for each A union N belong to F

the completion of (omega, sigma(F0), mju_star) is (omega, H, mju_star)

Let C be the collection of subsets of omega
C is called a Monotone Class if:
An in C s.t. An increasing or decreasing to A and A belong to C

M(C): the minimal monotone class on omega which containing C

Monotone Class Theorem:
Let F be a field, Let C be a monotone class
If C containing F then C containing sigma(F) and M(F)=sigma(F)

Carathe'odary Extension Theorem:
Let mju is a measure on a filed F
Assume mju is sigma-finite on F, then mju has unique extension of measure on sigma(F)

sigma-finite on F:
exist An in F such that mju(An) is finite for all n and unions of An is equal to omega

Approximation Theorem:
(omega, F, mju) is a measure space
F0 is a field s.t. sigma(F0)=F
Assume mju is sigma-finite on F0

If A belong to F and mju(A) < infinite
then exist B in F0 s.t. mju(A delta B)< e

看來很閒....
我只有一個疑問....mju是瞎咪東東

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