Note on set theory - Real Analysis
Notes on Lebesgue measure and integration
Set $X$ is a collection of objects. Class $C$ is a set of sets ("2nd-order set"). Space $(X, \cdots)$ is a set with certain structure.
countable set; equivalence of sets; power of a set; power of a countable set; power of the continuum
union; intersection; complement, relative complement/set difference, absolute complement; symmetric difference
The Cartesian product or direct product $A \times B$ of two sets $A, B$ is the set of all tuples $(a,b)$ composed of elements of the two sets: $A \times B = \{ (a,b) \mid a \in A, b \in B \}$.
inclusion; equality
Function or mapping (in short, map) $f: X \to Y$ from set $X$ to set $Y$ is a set of ordered pairs $(x,y)$ such that $\forall x \in X$, $\exists! y \in Y$: $(x,y) \in f$; in other definitions, the set $f$ is also called the graph of the function, and $f(x) = y$ is equivalent notation for $(x, y) \in f$. Transformation $u: X \to X$ is a mapping of a set to itself. Operator $A: X \to Y$ is a function/mapping from a space $X$ to another space $Y$, especially when $X$ and $Y$ are both vector spaces; the operation commonly denoted as $Ax = y$. Functional is an operator from a vector space $X$ of functions to a scalar field $\mathbb{F}$, usually $\mathbb{R}$ or $\mathbb{C}$. Correspondence or binary relation $(R, A, B)$ between two sets $A$ and $B$ is any subset of the Cartesian product $A \times B$. Morphism $\alpha \in H_{\mathcal{K}}(A, B)$ of a category $\mathcal{K}$ is a mapping from an object $A$ in the category into another $B$.
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 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.
A measure space is a measurable space possessing a measure: $(X, \Sigma, \mu)$.
A complete measure space is a measure space $(X, \Sigma, \mu)$ in which every subset of every null set is measurable: $S \subseteq N \in \Sigma, \mu(N) = 0 \Rightarrow S \in \Sigma$. It follows that those sets have measure zero.
The completion of a measure space is the smallest extension $(X, \Sigma_0, \mu_0)$ of the measure space $(X, \Sigma, \mu)$ such that the former is complete:
Lebesgue measure $\lambda$ (or $n$-volume) is the completion of the Borel measure on $\mathbb{R}^n$ (interval lengths and their products).
Lebesgue measure space $(\mathbb{R}^n, \mathcal{L}, \lambda)$ is the $n$-product of real numbers associated with the Lebesgue measure and all its Lebesgue measurable subsets.
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.
Haar measure, Hausdorff measure
Fubini Theorem establishes a connection between multiple integral and iterated integrals.
Theorem (Fubini): Given $(S,\mathcal{S},\mu)$ and $(T,\mathcal{T},\theta)$ are sigma-finite measure spaces, and $f: S\times T \to \mathbb{R}$ is a measurable function w.r.t. $(\mathcal{S} \times \mathcal{T}, \mathcal{B})$. If $f\geq 0$ or $\int_S \, \mathrm{d} \mu \left(\int_T |f| \,\mathrm{d} \theta \right) < \infty$, then:
$$\int_{S \times T} f \,\mathrm{d}(\mu \times \theta) = \int_S \, \mathrm{d} \mu \left(\int_T f \,\mathrm{d} \theta \right) = \int_T \,\mathrm{d} \theta \left( \int_S f \, \mathrm{d} \mu \right)$$