Note on set theory - Real Analysis

union,

intersection,

complement relative complement/set difference absolute complement

symmetric difference

The **cartesian product**, or **direct product**, of two sets $A, B$ is the set of all points $(a,b)$ where $a \in A, b \in B$:

$$A \times B = \{ (a,b) | a \in A, b \in B \}$$

inclusion,

equality

Countable set

equivalence of sets

power of a set,

power of a countable set,

power of the continuum

An **algebra over a set** is a class closed under all finite set operations.

$\mathcal{F}$ is an

algebra over set$X$ if it is:

- a non-empty subset of the power set $2^X$.
- closed under (absolute) complementation.
- closed under union and/or intersection.

Any set algebra $\mathcal{F}$ is a subalgebra of the power set Boolean Algebra of the underlying set $X$.

A **sigma-algebra** is an algebra closed under countable unions and/or intersections.

$\Sigma$ is a

$\sigma$-algebraof a set $X$ if it is:

- a non-empty subset of the power set $2^X$
- closed under (absolute) complementation
- closed under countable unions and/or intersections

The requirement of closure under countable unions is equivalent to closure under countable intersections, guaranteed by De Morgan's laws.

**Smallest $\sigma$-algebra** containing a subset of the power set is the intersection of all sigma-algebras containing it.

Given $S \subset 2^X$, the

smallest sigma-algebracontaining $S$ is $\Sigma = \bigcap \{ \mathcal{F} : S \subset \mathcal{F}, \mathcal{F} \text{ is a sigma-algebra of } X \}$.

We also call the smallest sigma-algebra containing $S$ the **sigma-algebra generated by $S$**.

**Borel $\sigma$-algebra** $\mathcal{B}$ is defined as the smallest $\sigma$-algebra containing a topology.

A **field of sets** is a pair $\langle X, \mathcal{F} \rangle$ where X is a set and $\mathcal{F}$ is an algebra over X.
The word "field" in "field of sets" is not used with the meaning of "field" from field theory.

A field of sets $(X,\Sigma)$ is a **measurable space** if the underlying algebra is a sigma-algebra.

A **measure** is a non-negative function on a sigma-algebra that is countably additive.

Given a sigma-algebra $\Sigma$, function $\mu: \Sigma \to \mathbb{R}_{\ge 0}$ is a

measureif it satisfies:

- Non-negativity: $\mu(E)\geq 0, \forall E \in \Sigma$;
- Null empty set: $\mu(\emptyset) =0$;
- Countable additivity: $\forall E_i \in \Sigma, E_i \cap E_j = \emptyset, i \ne j, i, j \in \mathbb{N} : \mu\left(\bigcup_{i \in \mathbb{N}} E_i\right ) = \sum_{i \in \mathbb{N}} \mu\left(E_i\right)$

A **measure space** is a measurable space possessing a measure: $(X,\Sigma,\mu)$.

A **finite measure** is a measure that assigns the entire domain set a finite value.

A **$\sigma$-finite measure** is a measure where the entire domain set is a countable union of measurable sets with finite measure.

The **product sigma-algebra** $\mathcal{S}\times\mathcal{T}$ is defined as the smallest sigma-algebra of product space $S \times T$ containing all measurable rectangles.
A product set $A \times B$ is a **measurable rectangle** if $A \in \mathcal{S}, B \in \mathcal{T}$.

The **product measure** $\mu\times\theta$ is a measure on the product measurable space that satisfies

$$(\mu\times\theta)(A\times B) = \mu(A) \theta(B), \forall A \in \mathcal{S}, B \in \mathcal{T}$$

Such product measures always exist, guaranteed by HahnāKolmogorov theorem. If the constituent measure spaces are sigma-finite, then product measure is uniquely defined:

$$(\mu\times\theta)(Q) = \int_S \mathrm{d}\theta \int_T \mathbf{1}_Q(s,t) \mathrm{d} \mu = \int_T \mathrm{d} \mu \int_S \mathbf{1}_Q(s,t) \mathrm{d} \theta, \forall Q \in \mathcal{S}\times\mathcal{T}$$

A **product measure space** of two measure spaces $(S,\mathcal{S},\mu)$ and $(T,\mathcal{T},\theta)$ is denoted as $(S\times T, \mathcal{S}\times\mathcal{T}, \mu\times\theta)$.

Even if both constituent measure spaces are complete, their product measure space still might not be complete.
Denote the **completion of product measure space** as $(S \times T, (\mathcal{S}\times\mathcal{T})^{*} , \mu\times\theta)$, where the **extended product measure** can be defined in a similar fashion.

- Wikibooks: Measure Theory. https://en.wikibooks.org/wiki/Measure_Theory

š· Category=Analysis