Note on set theory - Real Analysis

Set operations



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 $a \in A, b \in B$:

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

Set relations




Countable set

equivalence of sets

power of a set,

power of a countable set,

power of the continuum

Set algebra

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:

  1. a non-empty subset of the power set $2^X$.
  2. closed under (absolute) complementation.
  3. 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 $\sigma$-algebra of a set $X$ if it is:

  1. a non-empty subset of the power set $2^X$
  2. closed under (absolute) complementation
  3. 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-algebra containing $S$ 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.

Field of sets

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.

Measure space

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 measure if it satisfies:

  1. Non-negativity: $\mu(E)\geq 0, \forall E \in \Sigma$;
  2. Null empty set: $\mu(\emptyset) =0$;
  3. 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.

Product measure space

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 Hahnā€“Kolmogorov 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.


šŸ· Category=Analysis