Note on set theory - Real Analysis
A set $X$ is a collection of objects.
A space $(X, \cdots)$ is a set with certain structure.
A class $C$ is a set of sets.
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 of subsets of the set that is closed under all finite set operations: for a set $X$, a non-empty subset $\mathcal{F}$ of the power set $2^X$ that is closed under (absolute) complementation and closed under union and/or intersection.
Any set algebra $\mathcal{F}$ is a subalgebra of the power set of the underlying set $X$. (Boolean Algebra)
A sigma-algebra is an algebra closed under countable unions and/or intersections. A sigma-algebra $\Sigma$ of a set $X$ is:
By De Morgan's laws, closure under countable unions is equivalent to closure under countable intersections.
Smallest sigma-algebra containing a subset of the power set is the intersection of all sigma-algebras containing it. Given a class of subsets $S \subset 2^X$, the smallest sigma-algebra containing $S$ is $\Sigma = \bigcap \{ \mathcal{F} : S \subset \mathcal{F}, \mathcal{F} \text{ is a sigma-algebra of } X \}$. The smallest sigma-algebra containing $S$ is also called the sigma-algebra generated by $S$.
Borel sigma-algebra $\mathcal{B}$ is the smallest sigma-algebra containing a topology $T$.
A field of sets $(X, \mathcal{F})$ is a set $X$ with an algebra $\mathcal{F}$ over the set. 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 (测度, size) 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:
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 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 Hahn–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.