Notes on Lebesgue measure and integration

Measurable Space

Algebra of sets $(\mathcal{A}, (\complement, \cup, \cap))$ of a set $X$, a special case of Boolean algebra, is a non-empty class $\mathcal{A} \subset 2^X$ of its subsets that is closed under all finite set operations (absolute complement $\complement$, union $\cup$, and/or intersection $\cap$). Algebra of sets of a set $X$ is equivalent to ring of sets of $X$ that contains $X$. Subalgebra $\mathcal{A}'$ of an algebra $\mathcal{A}$ is an algebra such that $\mathcal{A}' \subset \mathcal{A}$. Any algebra of sets $\mathcal{A}$ of a set $X$ is a subalgebra of its power set $2^X$. Algebra generated by a class of subsets is the smallest algebra of sets $\mathcal{A}(\mathcal{S})$ of $X$ containing the class $\mathcal{S} \subset 2^X$: $\mathcal{A(S)} = \bigcap_{\mathcal{S} \subset \mathcal{A}_\alpha \subset 2^X} \mathcal{A}_\alpha$.

Sigma-algebra $(\Sigma, (\complement, \cup_{\mathbb{N}}, \cap_{\mathbb{N}}))$ of a set $X$ is an algebra of sets $\Sigma$ of $X$ that is closed under countable unions $\cup_{\mathbb{N}}$ and/or intersections $\cap_{\mathbb{N}}$. By De Morgan's laws, closure under countable unions and closure under countable intersections are equivalent. Sigma-algebra generated by a class of subsets is the smallest sigma-algebra $\Sigma(\mathcal{S})$ of $X$ containing the class $\mathcal{S} \subset 2^X$: $\Sigma(\mathcal{S}) = \bigcap_{\mathcal{S} \subset \Sigma_\alpha \subset 2^X} \Sigma_\alpha$. Borel sigma-algebra $\mathcal{B}$ is the sigma-algebra $\Sigma(\mathcal{T})$ generated by a topology $\mathcal{T}$; because of this, we also denote it as $\mathcal{B}(\mathcal{T})$.

Field of sets $(X, \mathcal{A})$ is a set $X$ with an algebra of sets $\mathcal{A}$ of the set. Here, "field" is not in the same sense of "field" in field theory. Measurable space $(X, \Sigma)$ is a field of sets where the algebra of sets $\Sigma$ is a sigma-algebra of $X$.

Measure Space

Measure (测度) $\mu: \Sigma \mapsto [0, \infty]$ on a sigma-algebra $\Sigma$ is a mapping from the sigma-algebra $\Sigma$ to extended nonnegative numbers $[0, \infty]$ that is countably additive: given $\{E_i\}_{i \in \mathbb{N}} \subset \Sigma$, if $E_i \cap E_j = \emptyset, \forall i \ne j$, then $\mu(\cup_{i \in \mathbb{N}} E_i) = \sum_{i \in \mathbb{N}} \mu(E_i)$. Finite measure is a measure that assigns the full set a finite value: $\mu(X) < \infty$. Sigma-finite measure is a measure where the full set is a countable union of measurable sets with finite measure: $X = \cup_{i \in \mathbb{N}} E_i$, $\mu(E_i) < \infty$.

Measure space $(X, \Sigma, \mu)$ is a measurable space $(X, \Sigma)$ with a measure $\mu$. Measure specifies the sizes of measurable sets of a measure space. Borel measure space $(X, \mathcal{T}, \mathcal{B(T)}, \mu)$ is a topological measure space with a set $X$, a topology $\mathcal{T}$, the Borel sigma-algebra $\mathcal{B(T)}$, and a measure $\mu$. Borel measure is the measure of a Borel measure space, i.e. any measure $\mu$ defined on the Borel sigma-algebra $\mathcal{B(T)}$. For the real numbers $\mathbb{R}$ with the usual topology $\mathcal{T}_d$, Borel measures that assign $\mu( (a,b] ) = b - a$ are called "the" Borel measure on $\mathbb{R}$.

Completion of a Measure Space

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.

Completion of a measure space is the smallest complete measure space $(X, \Sigma_0, \mu_0)$ that extends a measure space $(X, \Sigma, \mu)$:

1. Construct the class $Z$ of all subsets of measure zero subsets of $(X, \Sigma, \mu)$.
2. Generate the sigma-algebra $\Sigma_0$ from $\Sigma \cup Z$.
3. Extend the measure $\mu$ to $\mu_0$ such that $\forall C \in \Sigma_0$, $\mu_0 (C) = \inf_{C \subset D \in \Sigma} \mu(D)$.

Lebesgue measure space $(\mathbb{R}^n, \mathcal{L}, \lambda)$ is the completion of any Borel measure space $(\mathbb{R}^n, \mathcal{T_d}, \mathcal{B(T_d)}, \mu)$ where the measure of a product set of invervals is the product of interval lengths: the $n$-th Cartesian product of real numbers $\mathbb{R}^n$, all its Lebesgue measurable subsets $\mathcal{L}$, and the (outer) Lebesgue measure $\lambda$ (aka $n$-volume) $\lambda(A) = \inf\{\sum_{j\in\mathbb{N}} \lambda(I_j) : A \subset \cup_{j\in\mathbb{N}} I_j\}$.

Product Measure Space

Measurable rectangle of the product set $X \times Y$ of two measurable spaces $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ is a product set $A \times B$ of two measurable sets $A \in \Sigma_X$ and $B \in \Sigma_Y$. Product sigma-algebra $\Sigma_X \times \Sigma_Y$ of the product set $X \times Y$ of two measurable spaces $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ is the sigma-algebra generated by the class of measurable rectangles: $\Sigma_X \times \Sigma_Y = \Sigma(\mathcal{B})$, $\mathcal{B} = \{A \times B : A \in \Sigma_X, B \in \Sigma_Y\}$. Product measurable space $(X \times Y, \Sigma_X \times \Sigma_Y)$ of two measurable spaces $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ is the product set $X \times Y$ with the product sigma-algebra $\Sigma_X \times \Sigma_Y$.

Product measure $\mu \times \theta$ of two measure spaces $(X, \Sigma_X, \mu)$ and $(Y, \Sigma_Y, \theta)$ is a measure on the product measurable space $(X \times Y, \Sigma_X \times \Sigma_Y)$ such that $(\mu \times \theta)(A \times B) = \mu(A) \theta(B)$. By Hahn–Kolmogorov theorem, product measures always exist. If the constituent measure spaces are sigma-finite, then product measure is uniquely defined: $(\mu \times \theta)(Q) = \int_Y~\mathrm{d}\theta \int_X \mathbf{1}_Q(x, y)~\mathrm{d} \mu$ $= \int_X \mathrm{d} \mu \int_Y \mathbf{1}_Q(x, y) \mathrm{d} \theta$. Product measure space of two measure spaces $(X, \Sigma_X, \mu)$ and $(Y, \Sigma_Y, \theta)$ is the measure space $(X \times Y, \Sigma_X \times \Sigma_Y, \mu \times \theta)$. Product measure spaces might be incomplete, even if both constituent measure spaces are complete. Completion of a product measure space $(X \times Y, (\Sigma_X \times \Sigma_Y)^∗, (\mu \times \theta)^∗)$ can be defined in a similar fashion.

Integration

Lebesgue integration...

Integration on product space

Fubini Theorem establishes a connection between multiple integral and iterated integrals.

Theorem (Fubini): Given sigma-finite measure spaces $(S,\mathcal{S},\mu)$ and $(T,\mathcal{T},\theta)$, and a Borel-measurable function $f: S \times T \mapsto \mathbb{R}$. If $f\geq 0$ or $\int_S \, \mathrm{d} \mu \left(\int_T |f| \,\mathrm{d} \theta \right) < \infty$, then $\int_{S \times T} f \,\mathrm{d}(\mu \times \theta) = \int_S \, \mathrm{d} \mu \left(\int_T f \,\mathrm{d} \theta \right) = \int_T \,\mathrm{d} \theta \left( \int_S f \, \mathrm{d} \mu \right)$.

$L^p$ Space

L^p space $L^p (S, F, \mu)$ is a space of functions for which the $p$-th power of their absolute value is Lebesgue integrable.

Misc

Haar measure, Hausdorff measure