Notes on Lebesgue measure and integration
Algebra of sets or set algebra $(\mathcal{A}, (\cup, \cap; \complement))$ is an algebraic system consisting of a non-empty subset $\mathcal{A}$ of a power set $\mathcal{P}(X)$ and two operations: (1) union $\cup$ and/or intersection $\cap$ and (2) complement $\complement$ (such that the system is closed under any finite composition of operations). We call $X$ the underlying set of the set algebra. Any set algebra is a Boolean algebra. Algebra of sets is equivalent to ring of sets that includes the underlying set. Subalgebra $\mathcal{A}'$ of a set algebra is a subset of the set algebra that is also a set algebra: $\mathcal{A}' \subset \mathcal{A}$. The collection $\{\emptyset, X\}$ of the empty set and a set is a subalgebra of any set algebra based on the set, and thus is the smallest set algebra based on the set. Any set algebra is a subalgebra of the power set of its underlying set, and thus the largest set algebra based on a set is its power set. The union of members of any set algebra equals its underlying set: $\bigcup \mathcal{A} = X$. Algebra of sets $\mathcal{A(S)}$ generated by a class of subsets of a set is the smallest set algebra based on the set that contains the class: $\mathcal{A(S)} = \bigcap \{\mathcal{A : S \subset A \subset P}(X)\}$.
Sigma-algebra $(\Sigma, (\cup_{\mathbb{N}}, \cap_{\mathbb{N}}; \complement))$ is a set algebra that is closed under countable unions $\cup_{\mathbb{N}}$ and/or countable intersections $\cap_{\mathbb{N}}$. By De Morgan's laws, closure under countable unions and closure under countable intersections are equivalent. Sigma-algebra $\Sigma(\mathcal{S})$ generated by a subset of a power set is the smallest sigma-algebra based on the underlying set that contains the collection: $\Sigma(\mathcal{S}) = \bigcap \{\Sigma : \mathcal{S} \subset \Sigma \subset \mathcal{P}(X)\}$. Borel sigma-algebra $\mathcal{B(T)}$ is the sigma-algebra generated by a topology: $\mathcal{B(T)} = \Sigma(\mathcal{T})$.
Field of sets $(X, \mathcal{A})$ is a set and a set algebra based on the set. Here, "field" is not in the same sense of "field" in abstract algebra. Measurable space $(X, \Sigma)$ is a set endowed with a sigma-algebra. Measurable set in a measurable space is a set in its sigma-algebra. Borel measurable space $(X, \mathcal{T}, \mathcal{B(T)})$ is a topological space endowed with its Borel sigma-algebra. Borel set in a Borel measurable space is a set in its sigma-algebra.
Measurable mapping is a mapping between two measurable spaces such that the preimage of any measurable set is measurable: $f: X \mapsto Y$, $\{f^{-1}(B) : B \in \Sigma_Y\} \subset \Sigma_X$. I will use $\mathcal{M}(X, \Sigma_X; Y, \Sigma_Y)$ to denote the class of measurable mappings between two measurable spaces; if the sigma-algebras are Borel, and the topologies are unambiguous, the notation is simplified to $\mathcal{M}(X, Y)$. The class of measurable mappings is closed under composition. Every continuous mapping between Borel measurable spaces is measurable: $C(X, Y) \subset \mathcal{M}(X; Y)$. Measurable space isomorphism $f: (X, \Sigma_X) \cong (Y, \Sigma_Y)$ is a measurable mapping with a measurable inverse: $f \in \mathcal{M}(X, \Sigma_X; Y, \Sigma_Y)$, $f^{-1} \in \mathcal{M}(Y, \Sigma_Y; X, \Sigma_X)$. Measurable function is a measurable map to the Borel measurable space of real numbers with the usual topology. The class $\mathcal{M}(X, \Sigma)$ of measurable functions is closed under algebraic operations (addition, scalar and vector multiplications) and lattice operations (countable maximum and minimum).
Measure (测度) $\mu: \Sigma \mapsto [0, \infty]$ is a mapping from a sigma-algebra to the extended nonnegative real numbers which is distributive with countable union of mutually disjoint sets: $\mu(\sqcup_{i \in \mathbb{N}} A_i) = \sum_{i=1}^\infty \mu(A_i)$. The sum, converge or not, is independent of the order of the sets because the terms are nonnegative. Measure specifies the sizes of measurable sets of a measurable space. Inner measure $\mu_∗$ induced by a measure on a measurable space is the measure on the power set that maps each subset to the supremum of the measures of measureable subsets of the subset: $\forall A \in \mathcal{P}(X)$, $\mu_∗(A) = \sup\{\mu(B) : B \in [\emptyset, A] \cap \Sigma\}$. Outer measure $\mu^∗$ induced by a measure on a measurable space is the measure on the power set that maps each subset to the infimum of the measures of measureable subsets that include the subset: $\forall A \in \mathcal{P}(X)$, $\mu^∗(A) = \inf\{\mu(B) : B \in [A, X] \cap \Sigma\}$.
Finite measure is a measure that assigns the full set a finite value: $\mu(X) < \infty$. Normalized measure is a measure that assigns the full set the unit: $\mu(X) = 1$. Every finite measure can be normalized. Sigma-finite measure is a measure where the underlying set is a countable union of measurable sets with finite measure: $X = \cup_{i \in \mathbb{N}} A_i$, $\mu(A_i) < \infty$. Locally finite measure on a Borel measurable space is a measure such that every point has a neighborhood of finite measure.
Absolutely continuous measure $\nu \ll \mu$ w.r.t. a measure μ on a measurable space is a measure on the space such that any set of zero μ-measure is a set of zero ν-measure: $\mu(A) = 0$ then $\nu(A) = 0$. Absolutely continuous measure of a finite measure is equivalent to a measure whose value can be arbitrarily small if the corresponding μ-measure is sufficiently small: $\forall \varepsilon > 0$, $\exists \delta > 0$: $\mu^{-1}[0, \delta) \subset \nu^{-1}[0, \varepsilon)$. Singular measure $\nu \perp \mu$ w.r.t. a measure μ on a measurable space is a measure on the space such that there is a set of zero μ-measure whose complement is a set of zero ν-measure: $\exists A \in \Sigma$: $\mu(A) = 0$, $\nu(\complement A) = 0$. Radon-Nikodým Decomposition Theorem [@Radon1919; @Nikodým1930]: Every sigma-finite measure on a measurable space can be uniquely represented as the sum of an absolutely continuous measure and a singular measure, both w.r.t. another sigma-finite measure: $\exists \nu_a \ll \mu$, $\exists \nu_s \perp \mu$: $\nu = \nu_a + \nu_s$.
Measure space $(X, \Sigma, \mu)$ is a measurable space endowed with a measure. Probability space is a measurable space endowed with a normalized measure. Negligible subset in a measure space is a subset of zero measure: $\mu(A) = 0$. Full-measure subset in a measure space is a subset whose complement is negligible: $\mu(\complement A) = 0$. A property on a measure space holds almost everywhere (or a.e.) if it is a set of full measure: $\mu(\lnot P) = 0$. Almost equality $\approx_\mu$ or equality mod 0 on a measure space is a binary relation on its power set such that two subsets are almost equal if and only if their symmetric difference is negligible: $A \approx_\mu B$ iff $\mu(A \Delta B) = 0$.
Borel measure space $(X, \mathcal{T}, \mathcal{B(T)}, \mu)$ is a Borel measurable space endowed with a measure. Some authors call a measure on a Borel measurable space "Borel" if it satisfies certain properties. Borel measure space $(\mathbb{R}, \mathcal{T_d}, \mathcal{B(T_d)}, \mu)$ of real numbers is the real line endowed with the usual topology, the Borel sigma-algebra, and the measure that assigns each interval its length: $\mu([a, b]) = b - a$. Radon measure on a Borel measurable space with a Hausdorff topology is a locally finite measure such that the measure of a measurable set equals the supremum of the measures of its compact subspaces: $\mu(B) = \sup\{\mu(K) : K \subset B\}$. Every finite Radon measure on a Borel measurable space with a locally compact Hausdorff topology is outer regular, i.e. the measure of a measurable set equals the infimum of the measures of its neighborhoods: $\mu(B) = \inf\{\mu(U) : U \in [B, X] \cap \mathcal{T}\}$. Luzin Criterion for measurability of real functions [@Luzin1912]: A real function on an interval except for a set of zero measure is measurable if and only if it is continuous except for a set of arbitrarily small measure: $X \subset [a, b]$, $\mu(X) = b - a$, then $f \in \mathcal{M}(X, \mathcal{B})$ iff $\exists (A_n)_{n \in \mathbb{N}} \subset \mathcal{B}$, $X_n = \cup_{i=1}^n A_n$: $\lim_{n \to \infty} \mu(X_n) = b - a$, $f|_{X_n} \in C(X_n, Y)$. Luzin Criterion for measurability: A mapping from a finite Radon measure space to a second-countable Borel measurable space is measurable if and only if it is continuous except for an open subset of arbitrarily small measure: $f \in \mathcal{M}(X, \mathcal{B})$ iff $\exists (X_n)_{n \in \mathbb{N}} \subset \mathcal{T}_X^∗$: $\lim_{n \to \infty} \mu(X_n) = \mu(X)$, $f|_{X_n} \in C(X_n, Y)$.
Strict isomorphism or point isomorphism between two measure spaces is a measurable mapping with a measurable inverse and preserves the measures: $f: (X, \Sigma_X) \cong (Y, \Sigma_Y)$; $\forall A \in \Sigma_X$, $\mu(A) = \nu(f(A))$. Almost isomorphism or mod 0 isomorphism between two measure spaces is a strict isomorphism between some full measure subspaces.
μ-measurable subset in a measure space is a subset almost equal to a measurable set: $\exists B \in \Sigma$: $\mu(A \Delta B) = 0$. μ-measurable sigma-algebra $\Sigma_\mu$ w.r.t. a measure space is the sigma-algebra consisting of all the μ-measurable subsets in the space. Every μ-measurable sigma-algebra includes the original sigma-algebra: $\forall \mu$, $\Sigma \subset \Sigma_\mu$. Almost equality is an equivalence relation on a μ-measurable sigma-algebra. μ-measurable space $(X, \Sigma_\mu)$ w.r.t. a measure space is the measurable space consisting of the underlying set and the μ-measurable sigma-algebra. The inner and outer measures are the same on the μ-measurable space: $\mu_∗|\Sigma_\mu = \mu^∗|\Sigma_\mu$. Complete measure space is a measure space where every subset of every negligible set is measurable, i.e. negligible: $\cup_{\mu(N) = 0} \mathcal{P}(N) \subset \Sigma$; or equivalently, every μ-measurable set is measurable: $\Sigma_\mu = \Sigma$. Complete measure spaces of the same cardinality are strictly isomorphic if they are almost isomorphic and for every negligible set in one space there is a negligible set of the same cardinality in the other.
Completion $(X, \Sigma_\mu, \mu^∗)$ of a measure space is the complete measure space consisting of the underlying set, the μ-measurable sigma-algebra, and the outer or inner measure (considered as restricted to the μ-measurable sigma-algebra). The completion of a measure space is the smallest complete measure space that extends it: $\mu^∗|_\Sigma = \mu$, $\Sigma_{\mu^∗|_{\Sigma_\mu}} = \Sigma_\mu$, $\Sigma_\mu = \bigcap \{\Sigma' : \tilde \mu|_\Sigma = \mu, \Sigma_{\tilde \mu} = \Sigma'\}$. Lebesgue measure space $(\mathbb{R}, \mathcal{T_d}, \mathcal{L}, \lambda)$ of real numbers is the completion of the Borel measure space of real numbers: Lebesgue sigmal-algebra $\mathcal{L} = \Sigma_\mu$, Lebesgue measure $\lambda = \mu^∗|\mathcal{L}$. Standard probability space, Lebesgue–Rokhlin probability space, or Lebesgue space is a probability space isomorphic to a probability space consisting of an interval with the Lebesgue measure and a finite or countable set with a discrete measure. Ever separable complete metric space with the Borel sigma-algebra and a normalized measure completes to a Lebesgue space: $(X, \mathcal{B(T_d)}, \mu)$, then $(X,\Sigma_\mu,\mu^∗) \cong (I,\mathcal{L},\lambda) \sqcup (\mathbb{N},\mathcal{P}(\mathbb{N}),m)$.
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: $(\mu \times \theta)(Q) = \int_Y~\text{d}\theta \int_X \mathbf{1}_Q(x, y)~\text{d} \mu$ $= \int_X \text{d} \mu \int_Y \mathbf{1}_Q(x, y) \text{d} \theta$. 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 $(X \times Y, (\Sigma_X \times \Sigma_Y)_{\mu \times \theta}, (\mu \times \theta)^∗)$ can be defined in a similar fashion. Lebesgue measure space $(\mathbb{R}^n, \mathcal{T_d}, \mathcal{L}, \lambda)$ of real n-tuples is the completion of the n-th product measure space of the Borel measure space $(\mathbb{R}, \mathcal{T_d}, \mathcal{B(T_d)}, \mu)$ of real numbers. Lebesgue measure on the n-th Cartesian power of real numbers, aka n-volume: $\lambda(A) = \inf\{\sum_{i\in\mathbb{N}} \lambda(I_i) : A \subset \cup_{i\in\mathbb{N}} I_i\}$, where $I_i$ are n-dimensional intervals.
Product measure space of infinitely many normalized measure spaces is well defined.
Indefinite integral of a real function $f: (a, b) \mapsto \mathbb{R}$ on an open interval $(a, b)$ is the set of its primitives, i.e. functions whose derivatives equal $f$: $\int f~\text{d}x = \{F : \forall x \in (a, b), \text{d}F = f \text{d}x\}$. Definite integral of a real function $f: [a, b] \mapsto \mathbb{R}$ on an interval $[a, b]$ has a definition that evolved over time. Consider any countable partition $\{[x_n, x_{n+1}]\}_{n \in \mathbb{N}}$ of the domain $[a, b]$, specified by a sequence $\{\Delta x_n\}$ of non-negative reals with sum $\sum_n \Delta x_n = b-a$, such that $x_n = a + \sum_{i=1}^n \Delta x_i$. [@Cauchy1823] defined the definite integral as a limit of sums: $\int_a^b f~\text{d}x = \lim_{\max \Delta x_n \to 0} \sum_n f(x_n) \Delta x_n$. Continuous functions are Cauchy integrable. With arbitary evaluation points $\{\xi_n \in [x_n, x_{n+1}]\}_{n\in\mathbb{N}}$ given a partition, [@Riemann1853] defined the definite integral as the limit of Riemann sums: $\int_a^b f~\text{d}x = \lim_{\max \Delta x_n \to 0} \sum_n f(\xi_n) \Delta x_n$. A real function $f: [a, b] \mapsto \mathbb{R}$ is Riemann integrable iff it is bounded and it is countinous except for a zero-measure set of points [@Lebesgue1902]. Assuming $f$ is bounded on $[a, b]$, then there are sequences $\{m_n\}$ and $\{M_n\}$ where $m_n = \inf_{x_n \le x \le x_{n+1}} f(x)$ and $M_n = \sup_{x_n \le x \le x_{n+1}} f(x)$, [@Darboux1879] defined the definite integral as the limit of upper and lower Darboux sums: $\int_a^b f~\text{d}x = \lim_{\max \Delta x_n \to 0} \sum_n m_n \Delta x_n = \lim_{\max \Delta x_n \to 0} \sum_n M_n \Delta x_n$.
Fundamental theorem of calculus: given a continuous real function $f: [a, b] \mapsto \mathbb{R}$, (1) its definite integral equals the difference of its primitive's values at the interval ends, i.e. the Newton–Leibniz formula holds: $\int_a^b f~\text{d}x = F(b) - F(a)$; (2) its indefinite integral can be written as the definite integral with variable upper limit plus an arbitrary constant: $\int f~\text{d}x = \int_a^x f~\text{d}t + C$.
Stieltjes integral (or Riemann-Stieltjes integral) of a bounded real function $f: [a, b] \mapsto \mathbb{R}$ w.r.t. another bounded real function $G$ is the limit of Stieltjes sums [@Stieltjes1894]: $\int_a^b f~\text{d}G = \lim_{\max \Delta x_n \to 0} \sum_n f(\xi_n) \Delta G(x_n)$, where $\Delta G(x_n) = G(x_n) - G(x_{n-1})$. $f$ is called the integrand and $G$ the integrating function of the Stieltjes integral. Stieltjes integral generalizes Riemann integral, and if the integrating function $G$ has a Riemann integrable derivative $g$, Stieltjes integral reduces to Riemann integral: $\int_a^b f~\text{d}G = \int_a^b f g~\text{d}x$. Stieltjes integral is useful for curvilinear integral and the expectation of real random variables.
Simple function is a mapping $f: X \mapsto \{y_n\}_{n \in \mathbb{N}}$ from a measurable space $(X, \Sigma)$ to a countable set of real numbers such that the preimages are measurable: $\forall n \in \mathbb{N}$, $f^{-1}\{y_n\} \in \Sigma$. Lebesgue integral of a function $f: X \mapsto \mathbb{R}$ on a complete sigma-finite measure space $(X, \Sigma, \mu)$ is: (1) $\int_X f~\text{d}\mu = \sum_{n \in \mathbb{N}} y_n \mu(f^{-1}\{y_n\})$, if $f$ is a simple function and the series is absolutely convergent; (2) $\int_X f~\text{d}\mu = \lim_{n \to \infty} \int_X f_n~\text{d}\mu$, if there is a sequence $\{f_n\}$ of Lebesgue integrable simple functions that uniformly converges to $f$ almost everywhere; (3) $\int_X f~\text{d}\mu = \lim_{n \to \infty} \int_{A_n} f~\text{d}\mu$, if for any sequence $\{A_n\}$ of finite-measure sets successively expanding to $X$, the sequence of integrals converges [@Lebesgue1902]. A real-valued function is Lebesgue integrable if and only if its absolute value is Lebesgue integrable. The space of all Lebesgue integrable functions on a measure space is $L^1_\mu(X)$, see $L^p$ space. Lebesgue–Stieltjes integral generalizes the Lebesgue integral to measures of variable sign: if a measure of variable sign can be decomposed into the difference of two non-negative measures under which a function is Lebesgue integrable, then Lebesgue–Stieltjes integral of the function is the difference of the Lebesgue integrals; $\mu = \mu_1 - \mu_2$, $\int_X f~\text{d}\mu = \int_X f~\text{d}\mu_1 - \int_X f~\text{d}\mu_2$.
The integral concepts of Riemann, Stieltjes, and Lebesgue are very different: Riemann integral integrates a real function w.r.t. the volume (primitive of measure) of its domain $X$; Stieltjes integral integrates a real function w.r.t. a distribution $G: X \mapsto [0, 1]$ on its domain $X$; Lebesgue integral integrates a real-valued function w.r.t. a measure $\mu: \Sigma \mapsto [0, 1]$ on its domain $X$.
Lebesgue integrable functions are Lebesgue integrable on any measurable subdomain: $L^1_\mu(X) = \cap_{A \in \Sigma} L^1_\mu(A)$. Every Lebesgue integrable function $f \in L^1_\mu(X)$ on a complete sigma-finite measure space induces a measure $\mu_f = f \mu$ on the measurable space: $\forall A \in \Sigma$, $\mu_f(A) = \int_A f~\text{d}\mu$. Radon-Nikodym Theorem (representation of absolutely continuous measures): Any absolutely continuous measure w.r.t. the measure of a complete sigma-finite measure space can be uniquely represented as the product of a Lebesgue integrable function and the measure: $\forall \nu \ll \mu$, $\exists! f \in L^1_\mu(X)$: $\nu = f \mu$. Hence the set of all Lebesgue integrable functions on a complete sigma-finite measure space is isomorphic to the set of all absolutely continuous measures w.r.t. the measure: $L^1_\mu(X) \cong \{\nu: \nu \ll \mu\}$.
Passage to the Limit under the Lebesgue Integral [@Lebesgue1909]: If a sequence $\{f_n\}$ of measurable functions on $(X, \Sigma, \mu)$ that converges almost-everywhere to a function $f$ is absolutely bounded above by a Lebesgue integrable function $\Phi \in L^1(X, \Sigma, \mu)$, $\sup_{n \in \mathbb{N}} |f_n| \le \Phi$, then the sequence and the limit are all Lebesgue integrable and the limit of the sequence of integrals equals the integral of the limit: $\lim_{n \to \infty} \int_X f_n~\text{d}\mu = \int_X f~\text{d}\mu$.
Multiple Lebesgue integral is the Lebesgue integral of a multivariate function $f: \prod_i X_i \mapsto \mathbb{R}$, where the domain is the completion of the product measure space of complete sigma-finite measure spaces $\{(X_i, \Sigma_i, \mu_i)\}_{i=1}^n$. Multiple Integral as Repeated Integrals [@Fubini1907]: Given a measurable function $f: S \times T \mapsto \mathbb{R}$ on the product measure space $(S \times T, \mathcal{S \times T}, \mu \times \theta)$ of two sigma-finite measure spaces $(S, \mathcal{S}, \mu)$ and $(T, \mathcal{T}, \theta)$, if $f \geq 0$ or $\int_S \text{d} \mu \int_T |f| \,\text{d} \theta < \infty$, then $\int_{S \times T} f~\text{d}(\mu \times \theta) = \int_S \text{d} \mu \int_T f~\text{d} \theta = \int_T \text{d} \theta \int_S f~\text{d} \mu$.
L^p norm $\|\cdot\|_{p, \mu}$, where $p \in [1, \infty)$, of a measurable function on a complete sigma-finite measure space $(X, \Sigma, \mu)$ is the positive p-th root of the Lebesgue integral of the p-th power of the absolute function: $\|f\|_{p,\mu} = \sqrt[p]{\int_X |f|^p~\text{d} \mu}$. L^p metric $d_{p, \mu}(\cdot, \cdot)$ is the metric induced from an $L^p$ norm. The $L^2$ norm can be extended to an inner product of functions: $\langle f, g \rangle_\mu = \int_X f \bar g~\text{d} \mu$. L^p space $L^p_\mu(X)$ or $L^p(X, \Sigma, \mu)$ on a complete sigma-finite measure space is the $L^p$-normed space of measurable functions on the measure space: $L^p_\mu(X) = \{f : \int_X |f|^p~\text{d}\mu < \infty\}$. Note that the L in the name derives from Lebesgue, but do not confuse it with Lebesgue spaces. Every $L^p$ space, $p \in [1, \infty)$, is a Banach space. The $L^p$ space on a domain of integration in a Lebesgue measure space is the completion of the $L^p$-normed space of continuous real-valued functions with compact support on the domain: $L^p(D) = (\widehat C(D), \|\cdot\|_p)$. An $L^1$ space consists of all Lebesgue integrable functions on a measure space. Given a function in an $L^1$ space, the space also contains its absolute value, every function that differs only on a measure-zero subset, and every measurable function absolutely bounded by it: $f \in L^1_\mu(X)$ then $|f|, f + g, h \in L^1_\mu(X)$, where $\mu(\text{supp}(g)) = 0$ and $|h| \le |f|$. An $L^2$ space consists of all Lebesgue square integrable functions on a measure space, and is a Hilbert space when endowed with the inner product of functions. Equivalence $=$ of functions in an $L^p$ space is defined by almost equality: $f, g \in L^p_\mu(X)$, $f = g$ iff $f \approx_\mu g$.
Essential supremum $\text{ess} \sup |f|$ of a measurable function on a measure space $(X, \Sigma, \mu)$ is the smallest absolute bound of the function almost everywhere: $\text{ess} \sup |f| = \inf \{a : \mu(\{x: |f(x)| > a\}) = 0\}$. L^∞ norm $\|\cdot\|_{\infty, \mu}$ of a measurable function space is the norm that equals the essential supremum of the absolute function: $\|f\|_{\infty, \mu} = \text{ess} \sup_{x \in X} |f(x)|$. In comparison, uniform norm, sup norm, or infinity norm $\|\cdot\|_{\infty}$ on a scalar-valued function space is the norm that equals the supremum of the absolute function: $\|f\|_{\infty} = \sup_{x \in X} |f(x)|$. L^∞ space $L^\infty_\mu(X)$ or $L^\infty(X, \Sigma, \mu)$ on a complete sigma-finite measure space is the $L^\infty$-normed space of measurable functions on the measure space: $L^\infty_\mu(X) = \{f : \text{ess} \sup |f| < \infty \}$. Every $L^\infty$ space is a Banach space. The space of absolutely bounded, continuous real-valued functions on a domain of integration in a Lebesgue measure space, endowed with the uniform norm, is a Banach subspace of the $L^\infty$ space on the domain: $(C(D), \|\cdot\|_\infty) \subset L^\infty(D)$. $L^∞$ space is an extension of $L^p$ spaces with finite $p$, which has some different properties.
Inclusion of L^p and L^∞ spaces [@Villani1985]: Given a complete sigma-finite measure space $(X, \Sigma, \mu)$, the following are equivalent: (1) the measure is finite: $\mu(X) < \infty$; (2) one of its $L^p$ spaces includes another of a higher power: $\exists 1 \le p < q \le \infty$: $L^p_\mu(X) \supset L^q_\mu(X)$; (3) its $L^p$ spaces form a descending chain of set inclusion: $(\{L^p_\mu(X)\}_{p \in [1, \infty]}, \subset)$. And the following are also equivalent: (1) the measure is discrete: $\inf_{\mu(A) > 0} \mu(A) > 0$; (2) one of its $L^p$ spaces includes another of a lower power: $\exists 1 \le p < q \le \infty$: $L^p_\mu(X) \subset L^q_\mu(X)$; (3) its $L^p$ spaces form an ascending chain of set inclusion: $(\{L^p_\mu(X)\}_{p \in [1, \infty]}, \subset)$. As a result, almost-everywhere absolutely bounded measurable functions are Lebesgue integrable on subdomains of finite measure; $L^p$ spaces on a finite continuous measure space form a strictly descending chain of set inclusion; $L^p$ spaces on an infinite discrete measure space form a strictly ascending chain of set inclusion.
Integral operator $T_K$ between spaces of measurable functions on Hausdorff topological measure spaces, given a measurable function on the product measure space, is the operator defined by the integral w.r.t. the second variable of the function: $(X, \mathcal{T}_X, \mu)$, $(Y, \mathcal{T}_Y, \nu)$, $K: X \times Y \mapsto \mathbb{F}$, $g: Y \mapsto \mathbb{F}$, $T_K g: X \mapsto \mathbb{F}$, $(T_K g)(x) := \int_Y K(x, y) g(y) d \nu$. Kernel $K(x, y)$ of an integral operator is the measurable bivariate function in the integrand.
Boundedness of an Integral Operator:
Sobolev norm $\|\cdot\|_{s,p}$, where $s \in \mathbb{N}$ and $p \in [1, \infty)$, of a real-valued function on a domain of integration in a Lebesgue measure space is the sum of the $L^p$ norms of its partial derivatives up to the s-th order: $\|f\|_{s,p} = \sum_{|I| \le s} \left\|\frac{\partial f}{\partial x^I}\right\|_p$, where $I$ is a multi-index. Sobolev space $W^{s,p}(D)$ on a domain of integration in a Lebesgue measure space is the Sobolev-normed space of measureable functions on the domain: $W^{s,p}(D) = \{f: |I| \le s, \int_D \left|\frac{\partial f}{\partial x^I}\right|^p dx < \infty\}$, or in a shorter form with perhaps a little abuse of notation, $W^{s,p}(D) = \bigcap_{|I| \le s} \frac{\partial L^p(D)}{\partial x^I}$. Every Sobolev space is a Banach space. A Sobolev space is the completion of the Sobolev-normed space of smooth real-valued functions: $W^{s,p}(D) = (\widehat{C^\infty}(D), \|\cdot\|_{s,p})$.
Haar measure, Hausdorff measure.
improper integral. weak integral. strong integral.
Minimal L^p-metric $l_p$, $p \in [1, \infty)$, on the space of probability measures with p-th moments on a Borel measurable Euclidean space is the metric defined by the infimum of p-th root of p-th moments of their difference over all joint probabilities: $l_p(P, \tilde P) = \inf_\mu \|d(x, y)\|_{p, \mu}$, where $\mu(A \times \mathbb{R}^n) = P(A)$, $\mu(\mathbb{R}^n \times A) = \tilde P(A)$; more explicitly, $l_p(P, \tilde P) = \inf_\mu \sqrt[p]{\int_{\mathbb{R}^{2n}} \|x-y\|^p d\mu}$. The minimal $L^1$-metric is also called Kantorovich metric [@Kantorovich1940], Wasserstein metric [@Vaseršteĭn1969], or earth mover's distance [@Stolfi1994]. For probability distributions on the real line, the minimal $L^1$-metric equals the $L^1$-metric between CDFs: $l_1(P, \tilde P) = d_1(F, \tilde F)$, i.e. $l_1(P, \tilde P) = \int_\mathbb{R} |F(x) - \tilde F(x)| dx$. The minimal $L^2$-metric is also called Mallows metric [@Mallows1972]. The minimal $L^p$-metric is often called the p-th Wasserstein metric $W_p$.