Measurable Functions

Sigma Algebra

A family of subsets

ℱ \subseteq X

is said to be a

σ

-algebra on

X

whenever

$\emptyset, X \in ℱ$
If $A \in ℱ$ then $X ∖ A \in ℱ$
If $(A_{n})$ is a sequence of sets in $ℱ$ then

⋃_{n = 1}^{\infty} A_{n} \in ℱ

Measurable Space

An ordererd pair

(X, ℱ)

consisting of a set

X

and a sigma algebra on

X

is called a measurable space

Measurable With Resepect to a Sigma Algebra

Suppose that

(X, ℱ)

is a measure space, then any set

A \in ℱ

is said to be an

ℱ

-measurable set.

Whenever the sigma algebra is implicitly known we simplify the above and just say that $A$ is measurable.

Measure

A measure is an extended real-valued function

μ

defined on a

σ

-algebra

ℱ

of subsets of

X

such that

$μ (\emptyset) = 0$
$μ (E) \geq 0$ for all $E \in ℱ$
$μ$ is countably additive in the sense that if $(E_{n})$ is any disjoint sequence of sets in $ℱ$ , then

μ (⋃_{n = 1}^{\infty} E_{n}) = \sum_{n = 1}^{\infty} μ (E_{n}) .

Note that we say that a measure is finite or that it is $σ$ -finite when the the measure does not take on infinity in the above definition.

Measurable Function

Let

(X, ℱ)

be a measurable space, where

ℱ

is a

σ

-algebra of subsets of

X

, and let

(Y, 𝒢)

be another measurable space. A function

f : X \to Y

is called measurable if for every set

B \in 𝒢

we have:

f^{- 1} (B) \in ℱ for all B \in 𝒢 .

Almost Everywhere Convergence

Let

(X, ℱ, μ)

be a measure space, and let

{f_{n}}_{n = 1}^{\infty}

be a sequence of measurable functions

f_{n} : X \to ℝ

(or

ℂ

). We say that

{f_{n}}

converges to $f$ almost everywhere if there exists a set

N \subset X

with

μ (N) = 0

such that

\lim_{n \to \infty} f_{n} (x) = f (x) for all x \in X ∖ N .

In other words, the sequence ${f_{n} (x)}$ converges to $f (x)$ at all points $x$ except possibly on a set $N$ where the measure $μ (N)$ is zero. This means that the convergence holds "almost everywhere" in $X$ , despite there being potentially a small (measure-zero) set where it may fail to converge. Note that almost everywhere is often abbreviated as a.e.

ideas to be organized into the above

The notation $L_{μ}^{2} (X)$ represents the **space of square-integrable functions** on a measure space $(X, ℱ, μ)$ . Which are functions $f : X \to ℂ$ (or $ℝ$ ) such that: $\int_{X} | f (x) |^{2} d μ < \infty .$ This means that the squared absolute value of $f$ is integrable with respect to the measure $μ$ .

The differential $d μ$ in an integral $\int_{X} f d μ$ indicates that the integration is taken with respect to the measure $μ$ . The integral $\int_{X} f d μ$ is defined as a limit (or sum) over measurable "slices" of $X$ rather than just points. Formally, for a non-negative measurable function $f : X \to [0, \infty]$ , the integral with respect to $d μ$ is defined as $\int_{X} f d μ = \sup {\sum_{i = 1}^{n} f (x_{i}) μ (A_{i}) : {A_{i}}_{i = 1}^{n} is a finite measurable partition of X} .$ This expression takes a supremum over all possible finite partitions of $X$ into measurable sets $A_{i}$ with $x_{i} \in A_{i}$ and $\sum_{i = 1}^{n} f (x_{i}) μ (A_{i})$ providing an approximation of the integral over each slice weighted by $μ (A_{i})$ .