Abstract Measure Theory

27 minute read

posted: 01-Aug-2025 & updated: 03-Aug-2025

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Introduction

Preamble

Notations

sets of numbers
- $\naturals$ - set of natural numbers
- $\integers$ - set of integers
- $\integers_+$ - set of nonnegative integers
- $\rationals$ - set of rational numbers
- $\reals$ - set of real numbers
- $\preals$ - set of nonnegative real numbers
- $\ppreals$ - set of positive real numbers
- $\complexes$ - set of complex numbers
sequences $\seq{x_i}$ and the like
- finite $\seq{x_i}_{i=1}^n$, infinite $\seq{x_i}_{i=1}^\infty$ - use $\seq{x_i}$ whenever unambiguously understood
- similarly for other operations, e.g., $\sum x_i$, $\prod x_i$, $\cup A_i$, $\cap A_i$, $\bigtimes A_i$
- similarly for integrals, e.g., $\int f$ for $\int_{-\infty}^\infty f$
sets
- $\compl{A}$ - complement of $A$
- $A\sim B$ - $A\cap \compl{B}$
- $A\Delta B$ - $(A\cap \compl{B}) \cup (\compl{A} \cap B)$
- $\powerset(A)$ - set of all subsets of $A$
sets in metric vector spaces
- $\closure{A}$ - closure of set $A$
- $\interior{A}$ - interior of set $A$
- $\relint A$ - relative interior of set $A$
- $\boundary A$ - boundary of set $A$
set algebra
- $\sigma(\subsetset{A})$ - $\sigma$-algebra generated by $\subsetset{A}$, i.e., smallest $\sigma$-algebra containing $\subsetset{A}$
norms in $\reals^n$
- $\|x\|_p$ ($p\geq1$) - $p$-norm of $x\in\reals^n$, i.e., $(|x_1|^p + \cdots + |x_n|^p)^{1/p}$
- e.g., $\|x\|_2$ - Euclidean norm
matrices and vectors
- $a_{i}$ - $i$-th entry of vector $a$
- $A_{ij}$ - entry of matrix $A$ at position $(i,j)$, i.e., entry in $i$-th row and $j$-th column
- $\Tr(A)$ - trace of $A \in\reals^{n\times n}$, i.e., $A_{1,1}+ \cdots + A_{n,n}$
symmetric, positive definite, and positive semi-definite matrices
- $\symset{n}\subset \reals^{n\times n}$ - set of symmetric matrices
- $\possemidefset{n}\subset \symset{n}$ - set of positive semi-definite matrices; $A\succeq0 \Leftrightarrow A \in \possemidefset{n}$
- $\posdefset{n}\subset \symset{n}$ - set of positive definite matrices; $A\succ0 \Leftrightarrow A \in \posdefset{n}$
sometimes, use Python script-like notations (with serious abuse of mathematical notations)
- use $f:\reals\to\reals$ as if it were $f:\reals^n \to \reals^n$, e.g., $$ \exp(x) = (\exp(x_1), \ldots, \exp(x_n)) \quad \mbox{for } x\in\reals^n $$ and $$ \log(x) = (\log(x_1), \ldots, \log(x_n)) \quad \mbox{for } x\in\ppreals^n $$ which corresponds to Python code numpy.exp(x) or numpy.log(x) where x is instance of numpy.ndarray, i.e., numpy array
- use $\sum x$ to mean $\ones^T x$ for $x\in\reals^n$, i.e. $$ \sum x = x_1 + \cdots + x_n $$ which corresponds to Python code x.sum() where x is numpy array
- use $x/y$ for $x,y\in\reals^n$ to mean $$ \rowvecthree{x_1/y_1}{\cdots}{x_n/y_n}^T $$ which corresponds to Python code x / y where x and y are $1$-d numpy arrays
- use $X/Y$ for $X,Y\in\reals^{m\times n}$ to mean $$ \begin{my-matrix}{cccc} X_{1,1}/Y_{1,1} & X_{1,2}/Y_{1,2} & \cdots & X_{1,n}/Y_{1,n} \\ X_{2,1}/Y_{2,1} & X_{2,2}/Y_{2,2} & \cdots & X_{2,n}/Y_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ X_{m,1}/Y_{m,1} & X_{m,2}/Y_{m,2} & \cdots & X_{m,n}/Y_{m,n} \end{my-matrix} $$ which corresponds to Python code X / Y where X and Y are $2$-d numpy arrays

Some definitions

statement $P_n$, said to happen infinitely often or i.o. if $$ \left( \forall N\in\naturals \right) \left( \exists n > N \right) \left( P_n \right) $$

statement $P(x)$, said to happen almost everywhere or a.e. or almost surely or a.s. (depending on context) associated with measure space $\meas{X}{\algB}{\mu}$ if $$ \mu \set{x}{P(x)} = 1 $$ or equivalently $$ \mu \set{x}{\sim P(x)} = 0 $$

Some conventions

(for some subjects) use following conventions
- $0\cdot \infty = \infty \cdot 0 = 0$
- $(\forall x\in\ppreals)(x\cdot \infty = \infty \cdot x = \infty)$
- $\infty \cdot \infty = \infty$

Real Analysis

Set Theory

Some principles

$$ P(1) \& [P(n\Rightarrow P(n+1)] \Rightarrow (\forall n \in \naturals)P(n) $$

each nonempty subset of $\naturals$ has a smallest element

for $f:X\to X$ and $a\in X$, exists unique infinite sequence $\langle x_n\rangle_{n=1}^\infty\subset X$ such that $$ x_1=a $$ and $$ \left( \forall n \in \naturals \right) \left( x_{n+1} = f(x_n) \right) $$

note that $\Leftrightarrow$ $\Rightarrow$

Some definitions for functions

for $f:X\to Y$

terms, map and function, exterchangeably used
$X$ and $Y$, called domain of $f$ and codomain of $f$ respectively
$\set{f(x)}{x\in X}$, called range of $f$
for $Z\subset Y$, $f^{-1}(Z) = \set{x\in X}{f(x)\in Z}\subset X$, called preimage or inverse image of $Z$ under $f$
for $y\in Y$, $f^{-1}(\{y\})$, called fiber of $f$ over $y$
$f$, called injective or injection or one-to-one if $\left( \forall x\neq v \in X \right) \left( f(x) \neq f(v) \right)$
$f$, called surjective or surjection or onto if $\left( \forall x \in X \right) \left( \exists y in Y \right) (y=f(x))$
$f$, called bijective or bijection if $f$ is both injective and surjective, in which case, $X$ and $Y$, said to be one-to-one correspondece or bijective correspondece
$g:Y\to X$, called left inverse if $g\circ f$ is identity function
$h:Y\to X$, called right inverse if $f\circ h$ is identity function

Some properties of functions

for $f:X\to Y$

$f$ is injective if and only if $f$ has left inverse
$f$ is surjective if and only if $f$ has right inverse
hence, $f$ is bijective if and only if $f$ has both left and right inverse because if $g$ and $h$ are left and right inverses respectively, $g = g \circ (f\circ h) = (g\circ f)\circ h = h$
if $|X|=|Y|<\infty$, $f$ is injective if and only if $f$ is surjective if and only if $f$ is bijective

Countability of sets

set $A$ is countable if range of some function whose domain is $\naturals$
$\naturals$, $\integers$, $\rationals$: countable
$\reals$: not countable

Limit sets

for sequence, $\seq{A_n}$, of subsets of $X$
- limit superior or limsup of \seq{A_n}, defined by $$ \limsup \seq{A_n} = \bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_m $$
- limit inferior or liminf of \seq{A_n}, defined by $$ \liminf \seq{A_n} = \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty A_m $$
always $$ \liminf \seq{A_n} \subset \limsup \seq{A_n} $$
when $\liminf \seq{A_n} = \limsup \seq{A_n}$, sequence, $\seq{A_n}$, said to converge to it, denote $$ \lim \seq{A_n} = \liminf \seq{A_n} = \limsup \seq{A_n} = A $$

Algebras of sets

collection $\alg$ of subsets of $X$ called algebra or Boolean algebra if $$ (\forall A, B \in \alg) (A\cup B\in\alg) \mbox{ and } (\forall A \in \alg) (\compl{A}\in\alg) $$
- $(\forall A_1, \ldots, A_n \in \alg)(\cup_{i=1}^n A_i \in \alg)$
- $(\forall A_1, \ldots, A_n \in \alg)(\cap_{i=1}^n A_i \in \alg)$
algebra $\alg$ called $\sigma$-algebra or Borel field if
- every union of a countable collection of sets in $\alg$ is in $\alg$, i.e., $$ (\forall \seq{A_i})(\cup_{i=1}^\infty A_i \in \alg) $$
given sequence of sets in algebra $\alg$, $\seq{A_i}$, exists disjoint sequence, $\seq{B_i}$ such that $$ B_i \subset A_i \mbox{ and } \bigcup_{i=1}^\infty B_i = \bigcup_{i=1}^\infty A_i $$

Algebras generated by subsets

algebra generated by collection of subsets of $X$, $\coll$, can be found by $$ \alg = \bigcap \set{\algk{B}}{\algk{B} \in \collF} $$ where $\collF$ is family of all algebras containing $\coll$
- smallest algebra $\alg$ containing $\coll$, i.e., $$ (\forall \algk{B} \in \collF)(\alg \subset \algk{B}) $$
$\sigma$-algebra generated by collection of subsets of $X$, $\coll$, can be found by $$ \alg= \bigcap \set{\algk{B}}{\algk{B} \in \collG} $$ where $\collG$ is family of all $\sigma$-algebras containing $\coll$
- smallest $\sigma$-algebra $\alg$ containing $\coll$, i.e., $$ (\forall \algk{B} \in \collG)(\alg \subset \algk{B}) $$

Relation

$x$ said to stand in relation $\rel$ to $y$, denoted by $\relxy{x}{y}$
$\rel$ said to be relation on $X$ if $\relxy{x}{y}$ $\Rightarrow$ $x\in X$ and $y\in X$
$\rel$ is
- transitive if $\relxy{x}{y}$ and $\relxy{y}{z}$ $\Rightarrow$ $\relxy{x}{z}$
- symmetric if $\relxy{x}{y} = \relxy{y}{x}$
- reflexive if $\relxy{x}{x}$
- antisymmetric if $\relxy{x}{y}$ and $\relxy{y}{x}$ $\Rightarrow$ $x=y$
$\rel$ is
- equivalence relation if transitive, symmetric, and reflexive, e.g., modulo
- partial ordering if transitive and antisymmetric, e.g., “$\subset$''
- linear (or simple) ordering if transitive, antisymmetric, and $\relxy{x}{y}$ or $\relxy{y}{x}$ for all $x,y\in X$
  - e.g., “$\geq$'' linearly orders $\reals$ while “$\subset$'' does not $\powerset(X)$

Ordering

given partial order, $\prec$, $a$ is
- a first/smallest/least element if $x \neq a \Rightarrow a\prec x$
- a last/largest/greatest element if $x \neq a \Rightarrow x\prec a$
- a minimal element if $x \neq a \Rightarrow x \not\prec a$
- a maximal element if $x \neq a \Rightarrow a \not\prec x$
partial ordering $\prec$ is
- strict partial ordering if $x\not\prec x$
- reflexive partial ordering if $x\prec x$
strict linear ordering $<$ is
- well ordering for $X$ if every nonempty set contains a first element

Axiom of choice and equivalent principles

given a collection of nonempty sets, $\coll$, there exists $f:\coll\ \to \cup_{A\in\coll} A$ such that $$ \left( \forall A\in\coll\ \right) \left( f(A) \in A \right) $$

also called multiplicative axiom - preferred to be called to axiom of choice by Bertrand Russell for reason writte
no problem when $\coll$ is finite
need axiom of choice when $\coll$ is not finite

for particial ordering $\prec$ on $X$, exists a maximal linearly ordered subset $S\subset X$, i.e., $S$ is linearity ordered by $\prec$ and if $S\subset T\subset X$ and $T$ is linearly ordered by $\prec$, $S=T$

every set $X$ can be well ordered, i.e., there is a relation $<$ that well orders $X$

note that $\Leftrightarrow$ $\Leftrightarrow$

Infinite direct product

for collection of sets, $\seq{X_\lambda}$, with index set, $\Lambda$, $$ \bigtimes_{\lambda\in\Lambda} X_\lambda $$ called direct product

for $z=\seq{x_\lambda}\in\bigtimes X_\lambda$, $x_\lambda$ called $\lambda$-th coordinate of $z$

if one of $X_\lambda$ is empty, $\bigtimes X_\lambda$ is empty
axiom of choice is equivalent to converse, i.e., if none of $X_\lambda$ is empty, $\bigtimes X_\lambda$ is not empty if one of $X_\lambda$ is empty, $\bigtimes X_\lambda$ is empty
this is why Bertrand Russell prefers multiplicative axiom to axiom of choice for name of axiom ()

Real Number System

Field axioms

field axioms - for every $x,y,z\in\field$
- $(x+y)+z= x+(y+z)$ - additive associativity
- $(\exists 0\in\field)(\forall x\in\field)(x+0=x)$ - additive identity
- $(\forall x\in\field)(\exists w\in\field)(x+w=0)$ - additive inverse
- $x+y= y+x$ - additive commutativity
- $(xy)z= x(yz)$ - multiplicative associativity
- $(\exists 1\neq0\in\field)(\forall x\in\field)(x\cdot 1=x)$ - multiplicative identity
- $(\forall x\neq0\in\field)(\exists w\in\field)(xw=1)$ - multiplicative inverse
- $x(y+z) = xy + xz$ - distributivity
- $xy= yx$ - multiplicative commutativity
system (set with $+$ and $\cdot$) satisfying axiom of field called field
- e.g., field of module $p$ where $p$ is prime, $\primefield{p}$

Axioms of order

axioms of order - subset, $\field_{++}\subset \field$, of positive (real) numbers satisfies
- $x,y\in \field_{++} \Rightarrow x+y\in \field_{++}$
- $x,y\in \field_{++} \Rightarrow xy\in \field_{++}$
- $x\in \field_{++} \Rightarrow -x\not\in \field_{++}$
- $x\in \field \Rightarrow x=0\lor x\in \field_{++} \lor -x \in \field_{++}$
system satisfying field axioms & axioms of order called ordered field
- e.g., set of real numbers ($\reals$), set of rational numbers ($\rationals$)

Axiom of completeness

completeness axiom
- every nonempty set $S$ of real numbers which has an upper bound has a least upper bound, i.e., $$ \set{l}{(\forall x\in S)(l\leq x)} $$ has least element.
- use $\inf S$ and $\sup S$ for least and greatest element (when exist)
ordered field that is complete is complete ordered field
- e.g., $\reals$ (with $+$ and $\cdot$)
axiom of Archimedes
- given any $x\in\reals$, there is an integer $n$ such that $x<n$
corollary
- given any $x<y \in \reals$, exists $r\in\rationals$ such tat $x < r < y$

Sequences of $\reals$

sequence of $\reals$ denoted by $\seq{x_i}_{i=1}^\infty$ or $\seq{x_i}$
- mapping from $\naturals$ to $\reals$
limit of $\seq{x_n}$ denoted by $\lim_{n\to\infty} x_n$ or $\lim x_n$ - defined by $a\in\reals$ $$ (\forall \epsilon>0)(\exists N\in\naturals) (n \geq N \Rightarrow |x_n-a|<\epsilon) $$
- $\lim x_n$ unique if exists
$\seq{x_n}$ called Cauchy sequence if $$ (\forall \epsilon>0)(\exists N\in\naturals) (n,m \geq N \Rightarrow |x_n-x_m|<\epsilon) $$
Cauchy criterion - characterizing complete metric space (including $\reals$)
- sequence converges if and only if Cauchy sequence

Other limits

cluster point of $\seq{x_n}$ - defined by $c\in\reals$ $$ (\forall \epsilon>0, N\in\naturals)(\exists n>N)(|x_n-c|<\epsilon) $$
limit superior or limsup of $\seq{x_n}$ $$ \limsup x_n = \inf_n \sup_{k>n} x_k $$
limit inferior or liminf of $\seq{x_n}$ $$ \liminf x_n = \sup_n \inf_{k>n} x_k $$
$\liminf x_n \leq \limsup x_n$
$\seq{x_n}$ converges if and only if $\liminf x_n = \limsup x_n$ (=$\lim x_n$)

Open and closed sets

$O$ called open if $$ (\forall x\in O)(\exists \delta>0)(y\in\reals)(|y-x|<\delta\Rightarrow y\in O) $$
- intersection of finite collection of open sets is open
- union of any collection of open sets is open
$\closure{E}$ called closure of $E$ if $$ (\forall x \in \closure{E} \ \&\ \delta>0)(\exists y\in E)(|x-y|<\delta) $$
$F$ called closed if $$ F = \closure{F} $$
- union of finite collection of closed sets is closed
- intersection of any collection of closed sets is closed

Open and closed sets - facts

every open set is union of countable collection of disjoint open intervals
(Lindelöf) any collection $\coll$ of open sets has a countable subcollection $\seq{O_i}$ such that $$ \bigcup_{O\in\coll} O = \bigcup_{i} O_i $$
- equivalently, any collection $\collk{F}$ of closed sets has a countable subcollection $\seq{F_i}$ such that $$ \bigcap_{O\in\collk{F}} F = \bigcap_{i} F_i $$

Covering and Heine-Borel theorem

collection $\coll$ of sets called covering of $A$ if $$ A \subset \bigcup_{O\in\coll} O $$
- $\coll$ said to cover $A$
- $\coll$ called open covering if every $O\in\coll$ is open
- $\coll$ called finite covering if $\coll$ is finite
Heine-Borel theorem\index{Heine-Borel theorem}\index{Heine, Heinrich Eduard!Heine-Borel theorem}\index{Borel, Félix Édouard Justin Émile!Heine-Borel theorem} - for any closed and bounded set, every open covering has finite subcovering
corollary
- any collection $\coll$ of closed sets including at least one bounded set every finite subcollection of which has nonempty intersection has nonempty intersection.

Continuous functions

$f$ (with domain $D$) called continuous at $x$ if $$ (\forall\epsilon >0)(\exists \delta>0)(\forall y\in D)(|y-x|<\delta \Rightarrow |f(y)-f(x)|<\epsilon) $$
$f$ called continuous on $A\subset D$ if $f$ is continuous at every point in $A$
$f$ called uniformly continuous on $A\subset D$ if $$ (\forall\epsilon >0)(\exists \delta>0)(\forall x,y\in D)(|x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon) $$

Continuous functions - facts

$f$ is continuous if and only if for every open set $O$ (in co-domain), $f^{-1}(O)$ is open
$f$ continuous on closed and bounded set is uniformly continuous
extreme value theorem - $f$ continuous on closed and bounded set, $F$, is bounded on $F$ and assumes its maximum and minimum on $F$ $$ (\exists x_1, x_2 \in F)(\forall x\in F)(f(x_1) \leq f(x) \leq f(x_2)) $$
intermediate value theorem - for $f$ continuous on $[a,b]$ with $f(a) \leq f(b)$, $$ (\forall d)(f(a) \leq d \leq f(b))(\exists c\in[a,b])(f(c) = d) $$

Borel sets and Borel $\sigma$-algebra

Borel set
- any set that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement
Borel algebra or Borel $\sigma$-algebra
- smallest $\sigma$-algebra containing all open sets
- also
  - smallest $\sigma$-algebra containing all closed sets
  - smallest $\sigma$-algebra containing all open intervals (due to statement on page~)

Various Borel sets

countable union of closed sets (in $\reals$), called an $F_\sigma$ ($F$ for closed & $\sigma$ for sum)
- thus, every countable set, every closed set, every open interval, every open sets, is an $F_\sigma$ (note $(a,b)=\bigcup_{n=1}^\infty [a+1/n,b-1/n]$)
- countable union of sets in $F_\sigma$ again is an $F_\sigma$
countable intersection of open sets called a $G_\delta$ ($G$ for open & $\delta$ for durchschnitt - average in German)
- complement of $F_\sigma$ is a $G_\delta$ and vice versa
$F_\sigma$ and $G_\delta$ are simple types of Borel sets
countable intersection of $F_\sigma$'s is $F_{\sigma\delta}$, countable union of $F_{\sigma\delta}$'s is $F_{\sigma\delta\sigma}$, countable intersection of $F_{\sigma\delta\sigma}$'s is $F_{\sigma\delta\sigma\delta}$, etc., & likewise for $G_{\delta \sigma \ldots}$
below are all classes of Borel sets, but not every Borel set belongs to one of these classes $$ F_{\sigma}, F_{\sigma\delta}, F_{\sigma\delta\sigma}, F_{\sigma\delta\sigma\delta}, \ldots, G_{\delta}, G_{\delta\sigma}, G_{\delta\sigma\delta}, G_{\delta\sigma\delta\sigma}, \ldots, $$

Measure and Integration

Purpose of integration theory

purpose of “measure and integration'' slides
- abstract (out) most important properties of Lebesgue measure and Lebesgue integration
provide certain axioms that Lebesgue measure satisfies
base our integration theory on these axioms
hence, our theory valid for every system satisfying the axioms

Measurable space, measure, and measure space

family of subsets containing $\emptyset$ closed under countable union and completement, called $\sigma$-algebra
mapping of sets to extended real numbers, called set function
$\measu{X}{\algk{B}}$ with set, $X$, and $\sigma$-algebra of $X$, $\algk{B}$, called measurable space
- $A\in\algk{B}$, said to be measurable (with respect to \algk{B})
nonnegative set function, $\mu$, defined on $\algk{B}$ satisfying $\mu(\emptyset)=0$ and for every disjoint, $\seq{E_n}_{n=1}^\infty\subset \algk{B}$, $$ \mu\left(\bigcup E_n\right) = \sum \mu E_n $$ called measure on measurable space, $\measu{X}{\algk{B}}$
measurable space, $\measu{X}{\algk{B}}$, equipped with measure, $\mu$, called measure space and denoted by $\meas{X}{\algk{B}}{\mu}$

Measure space examples

$\meas{\reals}{\subsetset{M}}{\mu}$ with Lebesgue measurable sets, $\subsetset{M}$, and Lebesgue measure, $\mu$
$\meast{[0,1]}{\set{A\in\subsetset{M}}{A\subset[0,1]}}{\mu}$ with Lebesgue measurable sets, $\subsetset{M}$, and Lebesgue measure, $\mu$
$\meas{\reals}{\algB}{\mu}$ with class of Borel sets, $\algB$, and Lebesgue measure, $\mu$
$\meas{\reals}{\powerset(\reals)}{\mu_C}$ with set of all subsets of $\reals$, $\powerset(\reals)$, and counting measure, $\mu_C$
interesting (and bizarre) example
- $\meas{X}{\collk{A}}{\mu_B}$ with any uncountable set, $X$, family of either countable or complement of countable set, $\collk{A}$, and measure, $\mu_B$, such that $\mu_B A =0$ for countable $A\subset X$ and $\mu_B B=1$ for uncountable $B\subset X$

More properties of measures

for $A,B\in\algB$ with $A\subset B$ $$ \mu A \leq \mu B $$
for $\seq{E_n}\subset \algB$ with $\mu E_1 < \infty$ and $E_{n+1} \subset E_n$ $$ \mu\left(\bigcap E_n\right) = \lim \mu E_n $$
for $\seq{E_n}\subset \algB$ $$ \mu\left(\bigcup E_n\right) \leq \sum \mu E_n $$

Finite and $\sigma$-finite measures

measure, $\mu$, with $\mu(X)<\infty$, called finite
measure, $\mu$, with $X=\bigcup X_n$ for some $\seq{X_n}$ and $\mu(X_n)<\infty$, called $\sigma$-finite
- always can take $\seq{X_n}$ with disjoint $X_n$
Lebesgue measure on $[0,1]$ is finite
Lebesgue measure on $\reals$ is $\sigma$-finite
countering measure on uncountable set is not $\sigma$-measure

Sets of finite and $\sigma$-finite measure

set, $E\in \algB$, with $\mu E<\infty$, said to be of finite measure
set that is countable union of measurable sets of finite measure, said to be of $\sigma$-finite measure
measurable set contained in set of $\sigma$-finite measure, is of $\sigma$-finite measure
countable union of sets of $\sigma$-finite measure, is of $\sigma$-finite measure
when $\mu$ is $\sigma$-finite, every measurable set is of $\sigma$-finite

Semifinite measures

roughly speacking, nearly all familiar properties of Lebesgue measure and Lebesgue integration hold for arbitrary $\sigma$-finite measure
many treatment of abstract measure theory limit themselves to $\sigma$-finite measures
many parts of general theory, however, do not required assumption of $\sigma$-finiteness
undesirable to have development unnecessarily restrictive
measure, $\mu$, for which every measurable set of infinite measure contains measurable sets of arbitrarily large finite measure, said to be semifinite
every $\sigma$-finite measure is semifinite measure while measure, $\mu_B$, on page~ is not

Complete measure spaces

measure space, $\meas{X}{\algB}{\mu}$, for which $\algB$ contains all subsets of sets of measure zero, said to be complete, i.e., $$ (\forall B\in\algB \mbox{ with } \mu B=0) (A \subset B \Rightarrow A \in \algB) $$
- e.g., Lebesgue measure is complete, but Lebesgue measure restricted to $\sigma$-algebra of Borel sets is not
every measure space can be completed by addition of subsets of sets of measure zero
for $\meas{X}{\algB}{\mu}$, can find complete measure space $\meas{X}{\algB_0}{\mu_0}$ such that $$ \begin{eqnarray*} &-& \algB \subset \algB_0 \\ &-& E \in\algB \Rightarrow \mu E = \mu_0 E \\ &-& E \in\algB_0 \Leftrightarrow E = A \cup B \mbox{ where } B,C\in\algB, \mu C = 0, A\subset C \end{eqnarray*} $$
- $\meas{X}{\algB_0}{\mu_0}$ called completion of $\meas{X}{\algB}{\mu}$

Local measurability and saturatedness

for $\meas{X}{\algB}{\mu}$, $E\subset X$ for which $(\forall B\in\algB \mbox{ with }\mu B < \infty)(E\cap B\in\algB)$, said to be locally measurable
collection, $\algC$, of all locally measurable sets is $\sigma$-algebra containing $\algB$
measure for which every locally measurable set is measurable, said to be saturated
every $\sigma$-finite measure is saturated
measure can be extended to saturated measure, but (unlike completion) extension is not unique
- can take $\algC$ as extension for locally measurable sets, but measure can be extended on $\algC$ in more than one ways

Measurable functions

concept and properties of measurable functions in abstract measurable space almost identical with those of Lebesgue measurable functions (page~)
theorems and facts are essentially same as those of Lebesgue measurable functions
assume measurable space, $\measu{X}{\algB}$
for $f:X\to\ereals$, following are equivalent
- $(\forall a\in\reals) (\set{x\in X}{f(x) < a}\in\algB)$
- $(\forall a\in\reals) (\set{x\in X}{f(x) \leq a}\in\algB)$
- $(\forall a\in\reals) (\set{x\in X}{f(x) > a}\in\algB)$
- $(\forall a\in\reals) (\set{x\in X}{f(x) \geq a}\in\algB)$
$f:X\to\ereals$ for which any one of above four statements holds, called measurable or measurable with respect to \algB

Properties of measurable functions

for measurable functions, $f$ and $g$, and $c\in\reals$
- $f+c$, $cf$, $f+g$, $fg$, $f\vee g$ are measurable
for every measurable function sequence, $\seq{f_n}$
- $\sup f_n$, $\limsup f_n$, $\inf f_n$, $\liminf f_n$ are measurable
- thus, $\lim f_n$ is measurable if exists

Simple functions and other properties

$\varphi$ called simple function if for distinct $\seq{c_i}_{i=1}^n$ and measurable sets, $\seq{E_i}_{i=1}^n$ $$ \varphi(x) = \sum_{i=1}^n c_i \chi_{E_i}(x) $$
for nonnegative measurable function, $f$, exists nondecreasing sequence of simple functions, $\seq{\varphi_n}$, i.e., $\varphi_{n+1}\geq \varphi_n$ such that for every point in $X$ $$ f = \lim \varphi_n $$
- for $f$ defined on $\sigma$-finite measure space, we may choose $\seq{\varphi_n}$ so that every $\varphi_n$ vanishes outside set of finite measure
for complete measure, $\mu$, $f$ measurable and $f=g$ a.e. imply measurability of $g$

Define measurable function by ordinate sets

$\set{x}{f(x)<\alpha}$ sometimes called ordinate sets , which is nondecreasing in $\alpha$
below says when given nondecreasing ordinate sets, we can find $f$ satisfying $$ \set{x}{f(x)<\alpha} \subset B_\alpha \subset \set{x}{f(x)\leq\alpha} $$
for nondecreasing function, $h:D\to\algB$, for dense set of real numbers, $D$, i.e., $B_\alpha \subset B_\beta$ for all $\alpha<\beta$ where $B_\alpha = h(\alpha)$, exists unique measurable function, $f:X\to\ereals$ such that $f\leq \alpha$ on $B_\alpha$ and $f\geq \alpha$ on $X\sim B_\alpha$
can relax some conditions and make it a.e. version as below
for function, $h:D\to\algB$, for dense set of real numbers, $D$, such that $\mu(B_\alpha\sim B_\beta)=0$ for all $\alpha < \beta$ where $B_\alpha = h(\alpha)$, exists measurable function, $f:X\to\ereals$ such that $f\leq \alpha$ a.e. on $B_\alpha$ and $f\geq \alpha$ a.e. on $X\sim B_\alpha$
- if $g$ has the same property, $f=g$ a.e.

Integration

many definitions and proofs of Lebesgue integral depend only on properties of Lebesgue measure which are also true for arbitrary measure in abstract measure space (page~)
integral of nonnegative simple function, $\varphi(x) = \sum_{i=1}^n c_i \chi_{E_i}(x)$, on measurable set, $E$, defined by $$ \int_E \varphi d\mu= \sum_{i=1}^n c_i \mu (E_i \cap E) $$
- independent of representation of $\varphi$
for $a,b\in\ppreals$ and nonnegative simple functions, $\varphi$ and $\psi$ $$ \int (a\varphi + b\psi) = a \int\varphi + b \int\psi $$

Integral of bounded functions

for bounded function, $f$, identically zero outside measurable set of finite measure $$ \sup_{\varphi:\ \mathrm{simple},\ \varphi \leq f} \int \varphi = \inf_{\psi:\ \mathrm{simple},\ f \leq \psi} \int \psi $$ if and only if $f=g$ a.e. for measurable function, $g$
but, $f=g$ a.e. for measurable function, $g$, \iaoi\ $f$ is measurable with respect to completion of $\mu$, $\bar{\mu}$
natural class of functions to consider for integration theory are those measurable \wrt\ completion of $\mu$
thus, shall either assume $\mu$ is complete measure or define integral with respect to $\mu$ to be integral with respect to completion of $\mu$ depending on context unless otherwise specified

Difficulty of general integral of nonnegative functions

for Lebesgue integral of nonnegative functions (page~)
- first define integral for bounded measurable functions
- define integral of nonnegative function, $f$ as supremum of integrals of all bounded measurable functions, $h\leq f$, vanishing outside measurable set of finite measure
unfortunately, not work in case that measure is not semifinite
- e.g., if $\algB=\{\emptyset,X\}$ with $\mu \emptyset = 0$ and $\mu X = \infty$, we want $\int 1 d\mu=\infty$, but only bounded measurable function vanishing outside measurable set of finite measure is $h\equiv0$, hence, $\int g d\mu = 0$
to avoid this difficulty, we define integral of nonnegative measurable function directly in terms of integrals of nonnegative simple functions

Integral of nonnegative functions

for measurable function, $f:X\to\reals\cup\{\infty\}$, on measure space, $\meas{X}{\algB}{\mu}$, define integral of nonnegative extended real-valued measurable function $$ \int f d\mu = \sup_{\varphi:\ \mathrm{simple\ function},\ 0\leq \varphi\leq f} \int \varphi d\mu $$
however, definition of integral of nonnegative extended real-valued measurable function can be awkward to apply because
- taking supremum over large collection of simple functions
- not clear from definition that $\int(f+g) = \int f + \int g$
thus, first establish some convergence theorems, and determine value of $\int f$ as limit of $\int \varphi_n$ for increasing sequence, $\seq{\varphi_n}$, of simple functions converging to $f$

Fatou's lemma and monotone convergence theorem

Fatou's lemma - for nonnegative measurable function sequence, $\seq{f_n}$, with $\lim f_n = f$ a.e. on measurable set, $E$ $$ \int_E f \leq \liminf \int_E f_n $$
monotone convergence theorem - for nonnegative measurable function sequence, $\seq{f_n}$, with $f_n\leq f$ for all $n$ and with $\lim f_n = f$ a.e. $$ \int_E f = \lim \int_E f_n $$

Integrability of nonnegative functions

for nonnegative measurable functions, $f$ and $g$, and $a,b\in\preals$ $$ \int (af + bg) = a\int f + b\int g \mbox{ \& } \int f \geq 0 $$
- equality holds if and only if $f=0$ a.e.
monotone convergence theorem together with above yields, for nonnegative measurable function sequence, $\seq{f_n}$ $$ \int \sum f_n = \sum \int f_n $$
measurable nonnegative function, $f$, with $$ \int_E fd\mu <\infty $$ said to be integral (over measurable set, $E$, \wrt\ $\mu$)

Integral

arbitrary function, $f$, for which both $f^+$ and $f^-$ are integrable, said to be integrable
in this case, define integral $$ \int_E f = \int_E f^+ - \int_E f^- $$

Properties of integral

for $f$ and $g$ integrable on measure set, $E$, and $a,b\in\reals$
- $af+bg$ is integral and $$ \int_E (af+bg) = a \int_E f + b\int_E g $$
- if $|h|\leq |f|$ and $h$ is measurable, then $h$ is integrable
- if $f\geq g$ a.e. $$ \int f \geq \int g $$

Lebesgue convergence theorem

Lebesgue convergence theorem - for integral, $g$, over $E$ and sequence of measurable functions, $\seq{f_n}$, with $\lim f_n(x) = f(x)$ a.e. on $E$, if $$ |f_n(x)|\leq g(x) $$ then $$ \int_E f = \lim \int_E f_n $$

Setwise convergence of sequence of measures

preceding convergence theorems assume fixed measure, $\mu$
can generalize by allowing measure to vary
given measurable space, $\measu{X}{\algB}$, sequence of set functions, $\seq{\mu_n}$, defined on $\algB$, satisfying $$ (\forall E\in\algB) (\lim \mu_n E = \mu E) $$ for some set function, $\mu$, defined on $\algB$, said to converge setwise to $\mu$

General convergence theorems

generalization of Fatou's leamma - for measurable space, $\measu{X}{\algB}$, sequence of measures, $\seq{\mu_n}$, defined on $\algB$, converging setwise to $\mu$, defined on $\algB$, and sequence of nonnegative functions, $\seq{f_n}$, each measurable with respect to $\mu_n$, converging pointwise to function, $f$, measurable with respect to $\mu$ (compare with Fatou's lemma on page~) $$ \int f d\mu \leq \liminf\int f_n d\mu_n $$
generalization of Lebesgue convergence theorem - for measurable space, $\measu{X}{\algB}$, sequence of measures, $\seq{\mu_n}$, defined on $\algB$, converging setwise to $\mu$, defined on $\algB$, and sequences of functions, $\seq{f_n}$ and $\seq{g_n}$, each of $f_n$ and $g_n$, measurable with respect to $\mu_n$, converging pointwise to $f$ and $g$, measurable with respect to $\mu$, respectively, such that (compare with Lebesgue convergence theorem on page~) $$ \lim \int g_n d\mu_n = \int g d\mu < \infty $$ satisfy $$ \lim \int f_n d\mu_n = \int f\mu $$

$L^p$ spaces

for complete measure space, $\meas{X}{\algB}{\mu}$
- space of measurable functions on $X$ with with $\int |f|^p < \infty$, for which element equivalence is defined by being equal a.e., called $L^p$ spaces denoted by $L^p(\mu)$
- space of bounded measure functions, called $L^\infty$ space denoted by $L^\infty(\mu)$
norms
- for $p\in[1,\infty)$ $$ \|f\|_p=\left( \int |f|^p d\mu \right)^{1/p} $$
- for $p=\infty$ $$ \|f\|_\infty = \mathrm{ess\ sup} |f| = \inf \bigsetl{|g(x)|}{\mbox{measurable }g \mbox{ with } g=f \mbox{ a.e.}} $$
for $p\in[1,\infty]$, spaces, $L^p(\mu)$, are Banach spaces

Hölder's inequality and Littlewood's second principle

Hölder's inequality - for $p,q\in[1,\infty]$ with $1/p+1/q=1$, $f\in L^p(\mu)$ and $g\in L^q(\mu)$ satisfy $fg \in L^1(\mu)$ and $$ \|fg\|_1 = \int |fg| d\mu \leq \|f\|_p\|g\|_q $$
complete measure space version of Littlewood's second principle - for $p\in[1,\infty)$ $$ \begin{eqnarray*} &=& (\forall f\in L^p(\mu), \epsilon>0) \\ && (\exists \mbox{ simple function } \varphi \mbox{ vanishing outside set of finite measure}) \\ && \ \ \ \ \ \ \ (\|f-\varphi\|_p < \epsilon) \end{eqnarray*} $$

Riesz representation theorem

Riesz representation theorem - for $p\in[1,\infty)$ and bounded linear functional, $F$, on $L^p(\mu)$ and $\sigma$-finite measure, $\mu$, exists unique $g\in L^q(\mu)$ where $1/p+1/q=1$ such that $$ F(f) = \int fg d\mu $$ where $\|F\| = \|g\|_q$
if $p\in(1,\infty)$, Riesz representation theorem holds without assumption of $\sigma$-finiteness of measure

Measure and Outer Measure

General measures

consider some ways of defining measures on $\sigma$-algebra
recall that for Lebesgue measure
- define measure for open intervals
- define outer measure
- define notion of measurable sets
- finally derive Lebesgue measure
one can do similar things in general, e.g.,
- derive measure from outer measure
- derive outer measure from measure defined on algebra of sets

Outer measure

set function, $\mu^\ast:\powerset(X)\to[0,\infty]$, for space $X$, having following properties, called outer measure
- $\mu^\ast \emptyset = 0$
- $A\subset B \Rightarrow \mu^\ast A \leq \mu^\ast B$ (monotonicity)
- $E \subset \bigcup_{n=1}^\infty E_n \Rightarrow \mu^\ast E \leq \sum_{n=1}^\infty \mu^\ast E_n$ (countable subadditivity)
$\mu^\ast$ with $\mu^\ast X<\infty$ called finite
set $E\subset X$ satisfying following property, said to be measurable \wrt\ $\mu^\ast$ $$ (\forall A\subset X) (\mu^\ast(A) =\mu^\ast(A\cap E) + \mu^\ast(A\cap \compl{E})) $$
class, $\algB$, of $\mu^\ast$-measurable sets is $\sigma$-algebra
restriction of $\mu^\ast$ to $\algB$ is complete measure on $\algB$

Extension to measure from measure on an algebra

set function, $\mu:\alg\to[0,\infty]$, defined on algebra, $\alg$, having following properties, called measure on an algebra
- $\mu(\emptyset) = 0$
- $\left( \forall \mbox{ disjoint } \seq{A_n} \subset \alg \mbox{ with } \bigcup A_n \in \alg \right) \left( \mu\left(\bigcup A_n\right) = \sum \mu A_n \right)$
measure on an algebra, \alg, is measure if and only if $\alg$ is $\sigma$-algebra
can extend measure on an algebra to measure defined on $\sigma$-algebra, $\algB$, containing $\alg$, by
- constructing outer measure $\mu^\ast$ from $\mu$
- deriving desired extension $\bar{\mu}$ induced by $\mu^\ast$
process by which constructing $\mu^\ast$ from $\mu$ similar to constructing Lebesgue outer measure from lengths of intervals

Outer measure constructed from measure on an algebra

given measure, $\mu$, on an algebra, $\alg$
define set function, $\mu^\ast:\powerset(X)\to[0,\infty]$, by $$ \mu^\ast E = \inf_{\seq{A_n}\subset \alg,\ E\subset \bigcup A_n} \sum \mu A_n $$
$\mu^\ast$ called outer measure induced by $\mu$
then
for $A\in\alg$ and $\seq{A_n}\subset\alg$ with $A\subset \bigcup A_n$, $\mu A\leq \sum \mu A_n$
hence, $(\forall A\in\alg)(\mu^\ast A = \mu A)$
$\mu^\ast$ is outer measure
every $A\in\alg$ is measurable with respect to $\mu^\ast$

Regular outer measure

for algebra, $\alg$
- $\alg_\sigma$ denote sets that are countable unions of sets of $\alg$
- $\alg_{\sigma \delta}$ denote sets that are countable intersections of sets of $\alg_\sigma$
given measure, $\mu$, on an algebra, $\alg$ and outer measure, $\mu^\ast$ induced by $\mu$, for every $E\subset X$ and every $\epsilon>0$, exists $A\in\alg_\sigma$ and $B\in\alg_{\sigma \delta}$ with $E\subset A$ and $E\subset B$ $$ \mu^\ast A \leq \mu^\ast E + \epsilon \mbox{ and } \mu^\ast E = \mu^\ast B $$
outer measure, $\mu^\ast$, with below property, said to be regular $$ (\forall E\subset X, \epsilon>0) (\exists \mbox{ $\mu^\ast$-measurable set }A \mbox{ with } E\subset A) (\mu^\ast A \subset \mu^\ast E + \epsilon) $$
every outer measure induced by measure on an algebra is regular outer measure

Carathéodory theorem

given measure, $\mu$, on an algebra, $\alg$ and outer measure, $\mu^\ast$ induced by $\mu$
$E\subset X$ is $\mu^\ast$-measurable if and only if exist $A\in\alg_{\sigma\delta}$ and $B\subset X$ with $\mu^\ast B=0$ such that $$ E=A\sim B $$
- for $B\subset X$ with $\mu^\ast B=0$, exists $C\in\alg_{\sigma\delta}$ with $\mu^\ast C=0$ such that $B\subset C$
Carathéodory theorem - restriction, $\bar{\mu}$, of $\mu^\ast$ to $\mu^\ast$-measurable sets if extension of $\mu$ to $\sigma$-algebra containing $\alg$
- if $\mu$ is finite or $\sigma$-finite, so is $\bar{\mu}$ respectively
- if $\mu$ is $\sigma$-finite, $\bar{\mu}$ is only measure on smallest $\sigma$-algebra containing $\alg$ which is extension of $\mu$

Product measures

for countable disjoint collection of measurable rectangles, $\seq{(A_n \times B_n)}$, whose union is measurable rectangle, $A\times B$ $$ \lambda(A\times B) = \sum \lambda(A_n \times B_n) $$
for $x\in X$ and $E\in \algk{R}_{\sigma\delta}$ $$ E_x = \set{y}{\langle x,y\rangle \in E} $$ is measurable subset of $Y$
for $E\subset\algk{R}_{\sigma\delta}$ with $\mu \times \nu(E)<\infty$, function, $g$, defined by $$ g(x) = \nu E_x $$ is measurable function of $x$ and $$ \int g d\mu = \mu \times \nu(E) $$
XXX

Carathéodory outer measures

set, $X$, of points and set, $\Gamma$, of real-valued functions on $X$
two sets for which exist $a>b$ such that function, $\varphi$, greater than $a$ on one set and less than $b$ on the other set, said to be separated by function, $\varphi$
outer measure, $\mu^\ast$, with $(\forall A,B\subset X \mbox{ separated by } f\in\Gamma) (\mu^\ast(A\cup B) = \mu^\ast A + \mu^\ast B)$, called Carathéodory outer measure with respect to $\Gamma$
outer measure, $\mu^\ast$, on metric space, $\metrics{X}{\rho}$, for which $\mu^\ast(A\cup B)=\mu^\ast A + \mu^\ast B$ for $A,B\subset X$ with $\rho(A,B)>0$, called Carathéodory outer measure for $X$ or metric outer measure
for Carathéodory outer measure, $\mu^\ast$, with respect to $\Gamma$, every function in $\Gamma$ is $\mu^\ast$-measurable
for Carathéodory outer measure, $\mu^\ast$, for metric space, \metrics{X, \rho}, every closed set (hence every Borel set) is measurable with respect to $\mu^\ast$

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Sunghee Yun (He/Him)