Notes on Lebesgue measure and integration

**Algebra of sets** $(\mathcal{A}, (\cup, \cap, \complement))$ of a set $X$
is a non-empty subset $\mathcal{A} \subset \mathcal{P}(X)$ of its power set
that is closed under finite set operations union $\cup$ and/or intersection $\cap$,
and absolute complement $\complement$.
An algebra of sets is a Boolean algebra.
Any algebra of sets $\mathcal{A}$ of a set $X$ contains the set $X$ and the empty set $\emptyset$,
which in fact consist the smallest algebra of sets of $X$, and thus $\cup \mathcal{A} = X$.
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 $\mathcal{P}(X)$,
and thus $\mathcal{P}(X)$ is the largest algebra of sets of $X$.
**Algebra generated by a class of subsets** is
the smallest algebra of sets $\mathcal{A}(\mathcal{S})$ of $X$
containing the class $\mathcal{S} \subset \mathcal{P}(X)$:
$\mathcal{A(S)} := \cap_{\mathcal{S \subset A_\alpha \subset P}(X)} \mathcal{A_\alpha}$.

**Sigma-algebra** $(\Sigma, (\cup_{\mathbb{N}}, \cap_{\mathbb{N}}, \complement))$ 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 $\mathcal{S} \subset \mathcal{P}(X)$:
$\Sigma(\mathcal{S}) = \cap_{\mathcal{S} \subset \Sigma_\alpha \subset \mathcal{P}(X)} \Sigma_\alpha$.
**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 $\mathcal{B}(T)$.

**Field of sets** $(X, \mathcal{A})$ is a set $X$
with an algebra of sets $(\mathcal{A}, (\cup, \cap, \complement))$ of the set.
Here, "field" is not in the same sense of "field" in field theory.
**Measurable space** $(X, \Sigma)$ is a set $X$ with
a sigma-algebra of sets $(\Sigma, (\cup_{\mathbb{N}}, \cap_{\mathbb{N}}, \complement))$ of the set.
**Measurable set** in a measurable space is a set in its sigma-algebra.
**Borel set** is a measurable set in a Borel measurable space.

**Measurable mapping** is a mapping $f: X \mapsto Y$ between two measurable spaces
$(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ such that the preimage of any measurable set is measurable:
$\forall B \in \Sigma_Y, f^{-1}(B) \in \Sigma_X$.
The composition of measurable mappings is measurable.
The following theorems relate measurable mappings and continuous mappings.

Theorem: Every continuous mapping between two Borel measurable spaces is measurable.

Theorem (Luzin; $\mathcal{C}$-property of measurable mappings): A measurable mapping $f: X \mapsto Y$ from a Borel measure space $(X, \mathcal{T}, \mathcal{B(T)}, \mu)$ with a finite regular measure $\mu$ to a separable Borel measurable space $(Y, d, \mathcal{T}_d, \mathcal{B(T_d)})$ is continuous except for an open set of arbitrarily small measure: $\forall \varepsilon > 0, \exists A \in \mathcal{T}, \mu(A) < \varepsilon$ such that $f$ is continuous on $X \setminus A$.

**Measurable function** $f: X \mapsto \mathbb{R}$ refers to a measurable mapping to the real line
$\mathbb{R}$ with the Borel sigma-algebra $\mathcal{B(T_d)}$ of the usual topology.
The class of measurable functions is closed under arithmetical (linear combination, multiplication)
and lattice (countable max/min) operations.

**Measure** (测度) $\mu$ on a measurable space $(X, \Sigma)$
is a mapping $\mu: \Sigma \mapsto [0, \infty]$ from its sigma-algebra to extended nonnegative reals
and is countably additive (i.e. distributive with countable union of mutually disjoint sets):
if $\{A_i\}_{i \in \mathbb{N}} \subset \Sigma$, $\forall i \ne j, A_i \cap A_j = \emptyset$,
then $\mu(\cup_{i \in \mathbb{N}} A_i) = \sum_{i \in \mathbb{N}} \mu(A_i)$.
**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:
$X = \cup_{i \in \mathbb{N}} A_i$, $\mu(A_i) < \infty$.

**Absolutely continuous measure** w.r.t. a measure $\mu$ on a measurable space $(X, \Sigma)$
is a measure $\nu$ on the measurable space $(X, \Sigma)$ such that
any set of zero $\mu$-measure is a set of zero $\nu$-measure: $\mu(A) = 0 \implies \nu(A) = 0$,
denoted as $\nu \ll \mu$.
Absolutely continuous measure of a finite measure $\mu$ is equivalent to a measure
whose value can be arbitrarily small if the corresponding $\mu$-measure is sufficiently small:
$\forall \varepsilon > 0, \exists \delta > 0$: $\mu(A) < \delta \implies \nu(A) < \varepsilon$.
**Singular measure** w.r.t. a measure $\mu$ on a measurable space $(X, \Sigma)$
is a measure $\nu$ on the measurable space $(X, \Sigma)$ such that
there is a set $A$ of zero $\mu$-measure whose complement is a set of zero $\nu$-measure:
$\mu(A) = 0, \nu(\complement A) = 0$; denoted as $\nu \perp \mu$.

Theorem (Radon-Nikodým decomposition) [@Radon1919; @Nikodým1930]: Any sigma-finite measure $\nu$ on a sigma-finite measure space $(X, \Sigma, \mu)$ can be uniquely represented as the sum $\nu = \nu_a + \nu_s$ of an absolutely continuous measure $\nu_a$ and a singular measure $\nu_s$, both w.r.t. $\mu$.

**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 $\mathcal{B(T)}$.
**Borel regular measure** is a Borel measure such that $\forall A \in \mathcal{B(T)}$,
$\mu(A) = \inf_{A \subset B \in \mathcal{T}} \mu(B)$.
**Borel measure on the real line** $\mathbb{R}$ with the usual topology $\mathcal{T}_d$
is the Borel regular measure that assigns each interval its length: $\mu(a,b) = b - a$.
**Probability space** is equivalent to normalized measure space.

**Negligible set** in a measure space $(X, \Sigma, \mu)$ is a set $A$ of zero measure: $\mu(A) = 0$.
**Set of full measure** in a measure space $(X, \Sigma, \mu)$
is a set $A$ whose complement is negligible: $\mu(\complement A) = 0$.
A property $P$ on a measure space $(X, \Sigma, \mu)$ 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 $(X, \Sigma, \mu)$
is a binary relation on the power set $\mathcal{P}(X)$ such that two subsets $A, B \subset X$
are almost equal iff their symmetric difference is negligible:
$A \approx_\mu B \iff \mu(A \Delta B) = 0$.

**Inner measure** $\mu_∗$ and **outer measure** $\mu^∗$ induced by a measure $\mu$
on a measurable space $(X, \Sigma)$ are measures on the power set $\mathcal{P}(X)$ such that
$\mu_∗(A) := \sup_{[\emptyset, A] \cap \Sigma} \mu(B)$ and
$\mu^∗(A) := \inf_{[A, X] \cap \Sigma} \mu(B)$.

$\mu$**-measurable** set in a measure space $(X, \Sigma, \mu)$
is a set $A \subset X$ almost equal to a measurable set.
The class $\Sigma_\mu$ of all $\mu$-measurable sets in a measure space $(X, \Sigma, \mu)$
is a sigma-algebra with $\Sigma$ as a subalgebra.
Almost equality is an equivalence relation on $\Sigma_\mu$.
The induced inner and outer measures are the same on the $\mu$-measurable space $(X, \Sigma_\mu)$:
$\mu_∗|\Sigma_\mu = \mu^∗|\Sigma_\mu$.

**Complete measure space** is a measure space $(X, \Sigma, \mu)$
in which every subset of every null set is measurable: $\cup_{\mu(N) = 0} 2^N \subset \Sigma$;
equivalently, in which every $\mu$**-measurable** set is measurable: $\Sigma_\mu = \Sigma$.
Every subset of a negligible set in a complete measure space has measure zero.

**Completion** of a measure space $(X, \Sigma, \mu)$
is the smallest complete measure space that extends it.
It can be shown that the completion of measure space $(X, \Sigma, \mu)$
is the measure space $(X, \Sigma_\mu, \mu^∗|\Sigma_\mu)$
with sigma-algebra consisting of the $\mu$-measurable sets
and measure being the induced outer measure $\mu^∗$ (or inner measure $\mu_∗$)
restricted on the said sigma-algebra.

**Lebesgue measure space** $(\mathbb{R}, \mathcal{L}, \lambda)$ based on the real numbers
is the completion of the Borel measure space $(\mathbb{R}, \mathcal{T_d}, \mathcal{B(T_d)}, \mu)$
with the Borel measure $\mu$ on the real line:
**Lebesgue sigmal-algebra** $\mathcal{L} := \mathcal{B_\mu(T_d)}$;
**Lebesgue measure** $\lambda := \mu^∗|\mathcal{L}$.
**Lebesgue space** is a measure space isomorphic to a normalized measure space
$(\Delta \cup \{\alpha_i\}_{i \in \mathbb{N}}, \Sigma, \mu)$ consisting of
an interval $\Delta$ with a Lebesgue measure and at most a countable number of points
$\{\alpha_i\}_{i \in \mathbb{N}}$ with a discrete measure.
Any complete separable metric space $(X, d)$ with
the completion $(\mathcal{B_\mu(T_d)}, \mu^∗)$ of a normalized Borel measure is a Lebesgue 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**
$(X \times Y, (\Sigma_X \times \Sigma_Y)_{\mu \times \theta}, (\mu \times \theta)^∗)$
can be defined in a similar fashion.
**Lebesgue measure space** $(\mathbb{R}^n, \mathcal{L}, \lambda)$
based on the $n$-th Cartesian product of real numbers
is the completion of a product measure space of the Borel measure space
$(\mathbb{R}, \mathcal{T_d}, \mathcal{B(T_d)}, \mu)$.
The Lebesgue measure, 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 interval $(a, b)$
is the set of its primitives, i.e. functions whose derivatives equal $f$:
$\int f~\mathrm{d}x := \{F : \forall x \in (a, b), \mathrm{d}F = f \mathrm{d}x\}$.
**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:
$\int_a^b f~\mathrm{d}x := \lim_{\max \Delta x_n \to 0} \sum_n f(x_n) \Delta x_n$.
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:
$\int_a^b f~\mathrm{d}x := \lim_{\max \Delta x_n \to 0} \sum_n f(\xi_n) \Delta x_n$.
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:
$\int_a^b f~\mathrm{d}x := \lim_{\max \Delta x_n \to 0} \sum_n m_n \Delta x_n
= \lim_{\max \Delta x_n \to 0} \sum_n M_n \Delta x_n$.

**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: $\int_a^b f~\mathrm{d}x = F(b) - F(a)$;
(2) its indefinite integral can be written as the definite integral with variable upper limit
plus an arbitrary constant: $\int f~\mathrm{d}x = \int_a^x f~\mathrm{d}t + C$.

**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]:
$\int_a^b f~\mathrm{d}G := \lim_{\max \Delta x_n \to 0} \sum_n f(\xi_n) \Delta G(x_n)$,
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 generates Riemann integral,
and if the integrating function $G$ has a Riemann-integrable derivative $g$,
Stieltjes integral reduces to Riemann integral: $\int_a^b f~\mathrm{d}G = \int_a^b f g~\mathrm{d}x$.
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) $\int_X f~\mathrm{d}\mu := \sum_{n \in \mathbb{N}} y_n \mu(f^{-1}\{y_n\})$,
if $f$ is a simple function and the series is absolutely convergent;
(2) $\int_X f~\mathrm{d}\mu := \lim_{n \to \infty} \int_X f_n~\mathrm{d}\mu$,
if there is a sequence $\{f_n\}$ of Lebesgue integrable simple functions
that uniformly converges to $f$ almost everywhere;
(3) $\int_X f~\mathrm{d}\mu := \lim_{n \to \infty} \int_{A_n} f~\mathrm{d}\mu$,
if for any sequence $\{A_n\}$ of finite-measure sets successively expanding to $X$,
the sequence of integrals converges [@Lebesgue1902].
**Lebesgue–Stieltjes integral** generalizes the Lebesgue integral to measures of variable sign:
$\int_X f~\mathrm{d}\mu := \int_X f~\mathrm{d}\mu_1 - \int_X f~\mathrm{d}\mu_2$
if there are non-negative measures $\mu_1$ and $\mu_2$ under which $f$ is Lebesgue integrable.

The integral concepts of Riemann, Stieltjes, and Lebesgue are very different: Riemann integral integrates a real function w.r.t. the length/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: $L^1_\mu(X) = \cap_{A \in \Sigma} L^1_\mu(A)$. Lebesgue integral is countably additive: for all Lebesgue integrable function $f \in L^1_\mu(X)$ and for all countable class $\{A_i\}_{i \in \mathbb{N}} \subset \Sigma$ of mutually disjoint measurable sets, $\int_{\cup \{A_i\}_{i \in \mathbb{N}}} f~\mathrm{d}\mu = \sum_{i \in \mathbb{N}} \int_{A_i} f~\mathrm{d}\mu$. Thus, any Lebesgue integrable function $f \in L^1_\mu(X)$ on a complete sigma-finite measure space $(X, \Sigma, \mu)$ induces a measure $\mu_f(A) := \int_A f~\mathrm{d}\mu$ on the measurable space $(X, \Sigma)$. The set $L^1_\mu(X)$ of all Lebesgue integrable functions on a complete sigma-finite measure space $(X, \Sigma, \mu)$ and the set $\{\nu: \nu \ll \mu\}$ of measures absolutely continuous w.r.t. the measure $\mu$ are isomorphic: $\mu: L^1_\mu(X) \mapsto \{\nu: \nu \ll \mu\}$ (see the Radon–Nikodým theorem).

Theorem (Radon-Nikodym): Any measure $\nu$ absolutely continuous w.r.t. the measure $\mu$ of a complete sigma-finite measure space $(X, \Sigma, \mu)$ can be uniquely represented as $\nu = f \mu$ where $f \in L^1_\mu(X)$ is a Lebesgue integrable function: $\int_A ~\mathrm{d} \nu = \int_A f~\mathrm{d} \mu$.

Theorem (@Lebesgue1909; passage to the limit under the Lebesgue integral): 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: $\lim_{n \to \infty} \int_X f_n~\mathrm{d}\mu = \int_X f~\mathrm{d}\mu$.

**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 $\{(X_i, \Sigma_i, \mu_i)\}_{i=1}^n$.

Theorem (@Fubini1907; multiple integral as repeated integrals): 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 $\int_S \mathrm{d} \mu \int_T |f| \,\mathrm{d} \theta < \infty$, then $\int_{S \times T} f~\mathrm{d}(\mu \times \theta) = \int_S \mathrm{d} \mu \int_T f~\mathrm{d} \theta = \int_T \mathrm{d} \theta \int_S f~\mathrm{d} \mu$.

**Lp space** $L^p_\mu(X)$ or $L^p(X, \Sigma, \mu)$
on a complete sigma-finite measure space $(X, \Sigma, \mu)$, where $p \in [1, \infty)$, is
the set of functions on $X$ whose absolute value raised to the $p$-th power is Lebesgue integrable:
$L^p_\mu(X) := \{f : \int_X |f|^p~\mathrm{d}\mu < \infty\}$.
Note that the L in the name derives from Lebesgue, but do not confuse it with Lebesgue spaces.
**Equivalence** $=$ of two functions $f$ and $g$ in the Lp space $L^p_\mu(X)$
is defined by almost equality: $f = g \iff f \approx_\mu g$.
**Essential supremum** $\mathrm{ess} \sup |f|$ of a measurable function $f$
on a measure space $(X, \Sigma, \mu)$
is the smallest upper bound of the absolute value of the function almost everywhere:
$\mathrm{ess} \sup |f| := \inf \{a : \mu(\{x: |f(x)| > a\}) = 0\}$.
**L∞ space** $L^\infty_\mu(X)$ on a complete sigma-finite measure space $(X, \Sigma, \mu)$
consists of all the measurable functions on $X$ with a finite essential supremum:
$L^\infty_\mu(X) := \{f : \mathrm{ess} \sup |f| < \infty \}$.

Theorem (@Villani1985; inclusion of Lp spaces): 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 Lp 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 Lp 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 Lp 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 Lp spaces form an ascending chain of set inclusion: $(\{L^p_\mu(X)\}_{p \in [1, \infty]}, \subset)$.

As a result, almost-everywhere bounded measurable functions are Lebesgue integrable on subdomains of finite measure; Lp spaces on a finite continuous measure space form a strictly descending chain of set inclusion; Lp spaces on an infinite discrete measure space form a strictly ascending chain of set inclusion.

**Lp norm** $\|f\|_p := (\int_X |f|^p~\mathrm{d} \mu)^{1/p}$, where $p \in [1, \infty)$,
and $\|f\|_\infty := \mathrm{ess} \sup |f|$.
Discrete form ($X \subset \mathbb{N}$) of the Lp norm: $\|x\|_p = (\sum_i |x_i|^p)^{1/p}$.
Examples of discrete Lp norms:
L1 norm, $\|x\|_1 = \sum_i |x_i|$; L∞ norm, $\|x\|_\infty = \sup_i |x_i|$.
Any Lp space with the Lp norm, $p \in [1, \infty]$, is a complete normed space, i.e. Banach space.

$L^1_\mu(X)$ space consists of all Lebesgue integrable functions on $(X, \Sigma, \mu)$. $L^1_\mu(X)$ is closed under taking absolute value $|f|$ and perturbation $f + g$ on any measure-zero set $\mu(\mathrm{supp}(g))=0$, and contains any measurable function $|h| \le |f|$ absoluted bounded by one of its member functions.

$L^2_\mu(X)$ space consists of all Lebesgue square integrable functions on $(X, \Sigma, \mu)$. Any L2 space with inner product $\langle f, g \rangle = \int_X f g~\mathrm{d} \mu$ is a complete inner product space, i.e. a Hilbert space.

Haar measure, Hausdorff measure.

improper integral. weak integral. strong integral.