Finitely additive measure

$Finitely additive family$

F

is Finitely additive family of

X

:

1. ϕ \in F 2. A \in F \Rightarrow A^{c} \in F 3. A, B \in F \Rightarrow A \cup B \in F

X

is

F

-measurable :

X \in F

1. X \in F 2. A, B \in F \Rightarrow A \cap B, A - B \in F

K = {i = 1 ∐ n K_{i} ∣ n \in N, K_{i} = F_{i} \times G_{i} (F_{i} \in F, G_{i} \in G)}

K

: finitely additive family of

Z

m

is finitely additive measure on

F

:

1. m (ϕ) = 0 2. A, B \in F, A \cap B = ϕ \Rightarrow m (A + B) + m (A) + m (B)

1. A_{1}, A_{2}, ..., A_{n} \in F, A_{i} \cap A_{j} = ϕ (i \neq = j) \Rightarrow m (i = 1 ⋃ n A_{i}) = i = 1 \sum n m A_{i} 2. A, B \in F, B \subset A \Rightarrow m (A) \geq m (B) 2^{'} . A, B \in F, B \subset A, m (B) < \infty \Rightarrow m (A - B) = m (A) - m (B) 3. A_{1}, A_{2}, ..., A_{n} \in F \Rightarrow m (i = 1 ⋃ n A_{i}) \leq i = 1 \sum n m (A_{i})

m^{'}

is a complete finitely additive measure on

F

:

A = n = 1 ⋃ \infty A_{n} \in F \Rightarrow m (A) = n = 1 \sum \infty m (A_{n})

Box of

X

:

i = 1 ∐ i = n (a_{i}, b_{i}] (n \in N)

F_{N} := {Box of R^{N}}