Notes on Lebesgue measure and integration

**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** (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 $\mathbb{R}$.

**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 $(X, \Sigma, \mu)$:

- Construct the class $Z$ of all subsets of measure zero subsets of $(X, \Sigma, \mu)$.
- Generate the sigma-algebra $\Sigma_0$ from $\Sigma \cup Z$.
- 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**)
$\lambda(A) = \inf\{\sum_{j\in\mathbb{N}} \lambda(I_j) : A \subset \cup_{j\in\mathbb{N}} I_j\}$.

**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.

Lebesgue integration...

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

Haar measure, Hausdorff measure