Measurable Spaces And Functions

Sigma Algebra

Suppose that

X

is a set. A collection

𝒮 \subseteq 𝒫 (X)

is a $σ$ -algebra on

X

$\emptyset \in 𝒮$ ,
if $E \in 𝒮$ , then $X ∖ E \in 𝒮$ ,
if $E_{1}, E_{2}, \dots \in 𝒮$ , then $⋃_{k = 1}^{\infty} E_{k} \in 𝒮$ .

Sigma Algebras are Closed under Standard Set Operations

Suppose that

𝒮

is a

σ

-algebra on

X

. Then

$X \in 𝒮$ ,
if $D, E \in 𝒮$ , then $D \cup E \in 𝒮$ , $D \cap E \in 𝒮$ , and $D ∖ E \in 𝒮$ ,
if $E_{1}, E_{2}, \dots \in 𝒮$ , then $⋂_{k = 1}^{\infty} E_{k} \in 𝒮$ .

Measurable Space

A measurable space is an ordered pair

(X, 𝒮)

, where

X

is a set and

𝒮

is a

σ

-algebra on

X

Measurable Set

Suppose that

(X, 𝒮)

is a measurable space. A set

E \subseteq X

is $𝒮$ -measurable if

E \in 𝒮

Smallest Sigma Algebra Containing a Collection

Suppose that

X

is a set and

𝒜 \subseteq 𝒫 (X)

. The intersection of all

σ

-algebras on

X

that contain

𝒜

is a

σ

-algebra on

X

Borel

σ

-Algebra

The Borel $σ$ -algebra on

ℝ

is the smallest

σ

-algebra on

ℝ

that contains every open subset of

ℝ

Borel Set

A Borel set is an element of the Borel

σ

-algebra.

Inverse Image

Suppose that

f : X \to Y

is a function and

A \subseteq Y

. The inverse image of

A

under

f

f^{- 1} (A) : = {x \in X : f (x) \in A} .

Algebra of Inverse Images

Suppose that

f : X \to Y

. Then

$f^{- 1} (Y ∖ A) = X ∖ f^{- 1} (A)$ for every $A \subseteq Y$ ,
$f^{- 1} (⋃_{A \in 𝒜} A) = ⋃_{A \in 𝒜} f^{- 1} (A)$ for every collection $𝒜 \subseteq 𝒫 (Y)$ ,
$f^{- 1} (⋂_{A \in 𝒜} A) = ⋂_{A \in 𝒜} f^{- 1} (A)$ for every collection $𝒜 \subseteq 𝒫 (Y)$ .

Inverse Image of a Composition

Suppose that

f : X \to Y

g : Y \to W

, and

A \subseteq W

. Then

{(g \circ f)}^{- 1} (A) = f^{- 1} (g^{- 1} (A)) .

Measurable Function

Suppose that

(X, 𝒮)

is a measurable space. A function

f : X \to ℝ

is $𝒮$ -measurable if

f^{- 1} (B) \in 𝒮

for every Borel set

B \subseteq ℝ

Characteristic Function

Suppose that

E \subseteq X

. The characteristic function of

E

is the function

χ_{E} : X \to ℝ

defined by

χ_{E} (x) = {\begin{matrix} 1 & a m p; x \in E, \\ 0 & a m p; x \notin E . \end{matrix}

Ray Test for Measurability

Suppose that

(X, 𝒮)

is a measurable space and

f : X \to ℝ

. If

f^{- 1} ((a, \infty)) \in 𝒮

for every

a \in ℝ

, then

f

𝒮

-measurable.

Borel Measurable Function

Suppose that

X \subseteq ℝ

. A function

f : X \to ℝ

is Borel measurable if

f^{- 1} (B)

is a Borel set for every Borel set

B \subseteq ℝ

Continuous Functions are Borel Measurable

Every continuous real-valued function whose domain is a Borel subset of

ℝ

is Borel measurable.

Increasing Function

Suppose that

X \subseteq ℝ

and

f : X \to ℝ

. The function

f

is increasing if

f (x) \leq f (y)

whenever

x, y \in X

and

x < y

Strictly Increasing Function

Suppose that

X \subseteq ℝ

and

f : X \to ℝ

. The function

f

is strictly increasing if

f (x) < f (y)

whenever

x, y \in X

and

x < y

Increasing Functions are Borel Measurable

Every increasing real-valued function whose domain is a Borel subset of

ℝ

is Borel measurable.

Composition of Measurable Functions

Suppose that

(X, 𝒮)

is a measurable space and

f : X \to ℝ

𝒮

-measurable. If

g

is a Borel measurable real-valued function on a subset of

ℝ

that contains the range of

f

, then

g \circ f

𝒮

-measurable.

Algebraic Operations with Measurable Functions

Suppose that

(X, 𝒮)

is a measurable space and

f, g : X \to ℝ

are

𝒮

-measurable. Then

f + g

f - g

, and

f g

are

𝒮

-measurable. If

g (x) \neq 0

for every

x \in X

, then

f / g

𝒮

-measurable.

Pointwise Limit of Measurable Functions is Measurable

Suppose that

(X, 𝒮)

is a measurable space and

f_{1}, f_{2}, \dots : X \to ℝ

are

𝒮

-measurable. If

f (x) : = \lim_{k \to \infty} f_{k} (x)

exists for every

x \in X

, then

f

𝒮

-measurable.

Borel Subsets of the Extended Real Line

A subset

C \subseteq [- \infty, \infty]

is Borel if

C \cap ℝ

is a Borel subset of

ℝ

Extended Real-Valued Measurable Function

Suppose that

(X, 𝒮)

is a measurable space. A function

f : X \to [- \infty, \infty]

𝒮

-measurable if

f^{- 1} (B) \in 𝒮

for every Borel set

B \subseteq [- \infty, \infty]

Extended Ray Test for Measurability

Suppose that

(X, 𝒮)

is a measurable space and

f : X \to [- \infty, \infty]

. If

f^{- 1} ((a, \infty]) \in 𝒮

for every

a \in ℝ

, then

f

𝒮

-measurable.

Infimum and Supremum of Measurable Functions

Suppose that

(X, 𝒮)

is a measurable space and

f_{1}, f_{2}, \dots : X \to [- \infty, \infty]

are

𝒮

-measurable. Define

g (x) : = \inf {f_{k} (x) : k \in ℤ^{+}} and h (x) : = \sup {f_{k} (x) : k \in ℤ^{+}} .

Then

g

and

h

are

𝒮

-measurable.