Measure space

$Sigma algebra$

B

is

σ

-algebra of

X

:

1.\emptyset \in B 2. A \in B \Rightarrow A^{c} \in B 3. i \in N, A_{i} \in B \Rightarrow ⋃_{n = 1}^{\infty} A_{n} \in B

μ

is a measure on

B

:

1. μ (\emptyset) = 0 2.\forall n \in N (A_{n} \in B), A_{i} \cap A_{j} = \emptyset (i \neq = j) \Rightarrow μ (⋃_{n = 1}^{\infty} A_{n}) = \sum_{n = 1}^{\infty} μ (A_{n})

1. A, B \in F, A \subset B \Rightarrow μ (A) \leq μ (B) 1^{'} . A, B \in F, A \subset B, μ (A) < \infty \Rightarrow μ (B - A) = μ (B) - μ (A) 2. A_{n} \in F (n \in N) \Rightarrow μ (⋃_{n = 1}^{\infty} A_{n}) \leq \sum_{n = 1}^{\infty} μ (A_{n})

P

is true for almost everywhere in

E

:

μ (E_{0}) \neq = 0 \Rightarrow P (E_{0})

(X, B, μ)

is a measure space:

B

is

σ

-algebra on

X \land μ

is a measure on

F

M_{Γ}

is

σ

-algebra om

X

Γ

is a measure on

M_{Γ}

f

is measurable :

f^{- 1} (U)

is

B

-measurable