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 $\sigma$-algebra of 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-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.
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 measure if 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.
š· Category=Analysis