Measure Theory

25 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 $$ F_{\sigma}, F_{\sigma\delta}, F_{\sigma\delta\sigma}, F_{\sigma\delta\sigma\delta}, \ldots, G_{\delta}, G_{\delta\sigma}, G_{\delta\sigma\delta}, G_{\delta\sigma\delta\sigma}, \ldots, $$

Lebesgue Measure

Riemann integral

Riemann integral
- partition induced by sequence $\seq{x_i}_{i=1}^n$ with $a=x_1<\cdots<x_n=b$
- lower and upper sums
  - $L(f,\seq{x_i}) = \sum_{i=1}^{n-1} \inf_{x\in[x_i,x_{i+1}]} f(x) (x_{i+1}-x_{i})$
  - $U(f,\seq{x_i}) = \sum_{i=1}^{n-1} \sup_{x\in[x_i,x_{i+1}]} f(x) (x_{i+1}-x_{i})$
- always holds: $L(f,\seq{x_i}) \leq U(f,\seq{y_i})$, hence $$ \sup_{\seq{x_i}} L(f,\seq{x_i}) \leq \inf_{\seq{x_i}} U(f,\seq{x_i}) $$
- Riemann integrable if $$ \sup_{\seq{x_i}} L(f,\seq{x_i}) = \inf_{\seq{x_i}} U(f,\seq{x_i}) $$
every continuous function is Riemann integrable

Motivation - want measure better than Riemann integrable

consider indicator (or characteristic) function $\chi_\rationals:[0,1] \to [0,1]$ $$ \chi_\rationals(x) = \left\{\begin{array}{ll} 1 &\mbox{if } x \in \rationals \\ 0 &\mbox{if } x \not\in \rationals \end{array}\right. $$
not Riemann integrable: $\sup_{\seq{x_i}} L(f,\seq{x_i}) = 0 \neq 1 = \inf_{\seq{x_i}} U(f,\seq{x_i})$
however, irrational numbers infinitely more than rational numbers, hence
- want to have some integral $\int$ such that, e.g., $$ \int_{[0,1]} \chi_\rationals(x) dx = 0 \mbox{ and } \int_{[0,1]} (1-\chi_\rationals(x)) dx = 1 $$

Properties of desirable measure

want some measure $\mu:\subsetset{M}\to\preals=\set{x\in\reals}{x\geq0}$
- defined for every subset of $\reals$, i.e., $\subsetset{M} = \powerset(\reals)$
- equals to length for open interval $$ \mu[b,a] = b-a $$
- countable additivity: for disjoint $\seq{E_i}_{i=1}^\infty$ $$ \mu(\cup E_i) = \sum \mu(E_i) $$
- translation invariant $$ \mu(E+x) = \mu(E) \mbox{ for } x\in\reals $$
no such measure exists
not known whether measure with first three properties exists
want to find translation invariant countably additive measure
- hence, give up on first property

Race won by Henri Lebesgue in 1902!

mathematicians in 19th century struggle to solve this problem
race won by French mathematician, Henri Léon Lebesgue in 1902!
Lebesgue integral covers much wider range of functions
- indeed, $\chi_\rationals$ is Lebesgue integrable $$ \int_{[0,1]} \chi_\rationals(x) dx = 0 \mbox{ and } \int_{[0,1]} (1-\chi_\rationals(x)) dx = 1 $$

Outer measure

for $E\subset\reals$, define outer measure $\mu^\ast:\powerset(\reals)\to\preals$ $$ \mu^\ast E = \inf_{\seq{I_i}} \left\{\left.\sum l(I_i) \right| E\subset \cup I_i\right\} $$ where $I_i=(a_i,b_i)$ and $l(I_i) = b_i-a_i$
outer measure of open interval is length $$ \mu^\ast(a_i,b_i) = b_i-a_i $$
countable subadditivity $$ \mu^\ast\left(\cup E_i\right) \leq \sum \mu^\ast E_i $$
corollaries
- $\mu^\ast E = 0$ if $E$ is countable
- $[0,1]$ not countable

Measurable sets

$E\subset\reals$ called measurable if for every $A\subset\reals$ $$ \mu^\ast A = \mu^\ast (E\cup A) + \mu^\ast (\compl{E}\cup {A}) $$
$\mu^\ast E =0$, then $E$ measurable
every open interval $(a,b)$ with $a\geq -\infty$ and $b\leq \infty$ is measurable
disjoint countable union of measurable sets is measurable, i.e., $\cup E_i$ is measurable
collection of measurable sets is $\sigma$-algebra

Borel algebra is measurable

note
- every open set is disjoint countable union of open intervals (page~)
- disjoint countable union of measurable sets is measurable (page~)
- open intervals are measurable (page~)
hence, every open set is measurable
also
- collection of measurable sets is $\sigma$-algebra (page~)
- every open set is Borel set and Borel sets are $\sigma$-algebra (page~)
hence, Borel sets are measurable
specifically, Borel algebra (smallest $\sigma$-algebra containing all open sets) is measurable

Lebesgue measure

restriction of $\mu^\ast$ in collection $\subsetset{M}$ of measurable sets called Lebesgue measure $$ \mu:\subsetset{M}\to\preals $$
countable subadditivity - for $\seq{E_n}$ $$ \mu (\cup E_n) \leq \sum \mu E_n $$
countable additivity - for disjoint $\seq{E_n}$ $$ \mu (\cup E_n) = \sum \mu E_n $$
for dcreasing sequence of measurable sets, $\seq{E_n}$, i.e., $(\forall n\in\naturals)(E_{n+1} \subset E_n)$ $$ \mu\left( \bigcap E_n \right) = \lim \mu E_n $$

(Lebesgue) measurable sets are nice ones!

following statements are equivalent $$ \begin{eqnarray*} &-& E \mbox{ is measurable} \\ &-& (\forall \epsilon >0) (\exists \mbox{ open } O\supset E) (\mu^\ast(O\sim E)<\epsilon) \\ &-& (\forall \epsilon >0) (\exists \mbox{ closed } F\subset E) (\mu^\ast(E\sim F)<\epsilon) \\ &-& (\exists G_\delta) (G_\delta \supset E) (\mu^\ast(G_\delta\sim E)<\epsilon) \\ &-& (\exists F_\sigma) (F_\sigma \subset E) (\mu^\ast(E\sim F_\sigma)<\epsilon) \end{eqnarray*} $$
if $\mu^\ast E$ is finite, above statements are equivalent to $$ (\forall \epsilon>0) \left(\exists U = \bigcup_{i=1}^n (a_i,b_i) \right) (\mu^\ast (U\Delta E) < \epsilon) $$

Lebesgue measure resolves problem in movitation

let $$ E_1 = \set{x\in[0,1]}{x\in\rationals},\ E_2 = \set{x\in[0,1]}{x\not\in\rationals} $$
$\mu^\ast E_1=0$ because $E_1$ is countable, hence measurable and $$ \mu E_1 = \mu^\ast E_1 = 0 $$
algebra implies $E_2 = [0, 1] \cap \compl{E_1}$ is measurable
countable additivity implies $\mu E_1 + \mu E_2 = \mu[0,1] = 1$, hence $$ \mu E_1 = 1 $$

Lebesgue Measurable Functions

Lebesgue measurable functions

for $f:X\to\reals\cup\{-\infty, \infty\}$, i.e., extended real-valued function, the followings are equivalent
- for every $a\in\reals$, $\set{x\in{X}}{f(x) < a}$ is measurable
- for every $a\in\reals$, $\set{x\in{X}}{f(x) \leq a}$ is measurable
- for every $a\in\reals$, $\set{x\in{X}}{f(x) > a}$ is measurable
- for every $a\in\reals$, $\set{x\in{X}}{f(x) \geq a}$ is measurable
if so,
- for every $a\in\reals\cup\{-\infty, \infty\}$, $\set{x\in{X}}{f(x) = a}$ is measurable
extended real-valued function, $f$, called (Lebesgue) measurable function if
- domain is measurable
- any one of above four statements holds

Properties of Lebesgue measurable functions

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

Almost everywhere - a.e.

statement, $P(x)$, called almost everywhere or a.e. if $$ \mu \set{x}{\sim P(x)} = 0 $$
- e.g., $f$ said to be equal to $g$ a.e. if $\mu\set{x}{f(x)\neq g(x)}=0$
- e.g., $\seq{f_n}$ said to converge to $f$ a.e. if $$ (\exists E \mbox{ with } \mu E=0)(\forall x \not\in E)(\lim f_n (x) = f(x)) $$
facts
- if $f$ is measurable and $f=g$ i.e., then $g$ is measurable
- if measurable extended real-valued $f$ defined on $[a,b]$ with $f(x) \in\reals$ a.e., then for every $\epsilon>0$, exist step function $g$ and continuous function $h$ such that $$ \mu\set{x}{|f-g| \geq \epsilon} < \epsilon,\ \mu\set{x}{|f-h| \geq \epsilon} < \epsilon $$

Characteristic \& simple functions

for any $A\subset\reals$, $\chi_A$ called characteristic function if $$ \chi_A(x) = \left\{\begin{array}{ll} 1 & x\in A\\ 0 & x\not\in A\\ \end{array}\right. $$
- $\chi_A$ is measurable if and only if $A$ is measurable
measurable $\varphi$ called simple if for some distinct $\seq{a_i}_{i=1}^n$ $$ \varphi(x) = \sum_{i=1}^n a_i \chi_{A_i}(x) $$ where $A_i = \set{x}{x= a_i}$

Littlewood's three principles

let $M(E)$ with measurable set, $E$, denote set of measurable functions defined on $E$
every (measurable) set is nearly finite union of intervals, e.g.,
- $E$ is measurable if and only if $$ (\forall \epsilon>0) (\exists \{I_i: \mbox{open\ interval}\}_{i=1}^n) (\mu^\ast(E \Delta (\cup I_n)) < \epsilon) $$
every (measurable) function is nearly continuous, e.g.,
- (Lusin's theorem) $$ (\forall f \in M[a,b])(\forall \epsilon >0)(\exists g \in C[a,b]) (\mu\set{x}{f(x)\neq g(x)}< \epsilon) $$
every convergent (measurable) function sequence is nearly uniformly convergent, e.g., $$ \begin{eqnarray*} &=& (\forall \mbox{ measurable }\seq{f_n} \mbox{ converging to } f \mbox { a.e. on } E \mbox{ with } \mu E<\infty) \\ && (\forall \epsilon>0 \mbox{ and } \delta>0) (\exists A\subset E \mbox{ with } \mu(A)<\delta \mbox{ and } N\in\naturals) \\ && (\forall n > N, x\in E\sim A)(|f_n(x)-f(x)|<\epsilon) \end{eqnarray*} $$

Egoroff's theorem

Egoroff theorem - provides stronger version of third statement on page~ $$ \begin{eqnarray*} &=& (\forall \mbox{ measurable }\seq{f_n} \mbox{ converging to } f \mbox { a.e. on } E \mbox{ with } \mu E<\infty) \\ && (\exists A\subset E \mbox{ with } \mu(A)<\epsilon) (f_n \mbox{ uniformly converges to } f \mbox{ on } E\sim A ) \end{eqnarray*} $$

Lebesgue Integral

Integral of simple functions

canonical representation of simple function $$ \varphi(x) = \sum_{i=1}^n a_i \chi_{A_i}(x) $$ where $a_i$ are distinct $A_i=\{x|\varphi(x)=a_i\}$ - note $A_i$ are disjoint
when $\mu\set{x}{\varphi(x)\neq0}< \infty$ and $\varphi = \sum_{i=1}^n a_i \chi_{A_i}$ is canonical representation, define integral of $\varphi$ by $$ \int \varphi = \int \varphi (x) dx= \sum_{i=1}^n a_i \mu A_i $$
when $E$ is measurable, define $$ \int_E \varphi = \int \varphi \chi_E $$

Properties of integral of simple functions

for simple functions $\varphi$ and $\psi$ that vanish out of finite measure set, i.e., $\mu\set{x}{\varphi(x)\neq0}<\infty$, $\mu\set{x}{\psi(x)\neq0}<\infty$, and for every $a,b\in\reals$ $$ \int (a\varphi + b\psi) = a \int\varphi + b \int\psi $$
thus, even for simple function, $\varphi = \sum_{i=1}^n a_i \chi_{A_i}$ that vanishes out of finite measure set, not necessarily in canonical representation, $$ \int \varphi = \sum_{i=1}^n a_i \mu A_i $$
if $\varphi \geq \psi$ a.e. $$ \int \varphi \geq \int \psi $$

Lebesgue integral of bounded functions

for bounded function, $f$, and finite measurable set, $E$, $$ \sup_{\varphi:\ \mathrm{simple},\ \varphi \leq f} \int_E \varphi \leq \inf_{\psi:\ \mathrm{simple},\ f \leq \psi} \int_E \psi $$
- if $f$ is defined on $E$, $f$ is measurable function if and only if $$ \sup_{\varphi:\ \mathrm{simple},\ \varphi \leq f} \int_E \varphi = \inf_{\psi:\ \mathrm{simple},\ f \leq \psi} \int_E \psi $$
for bounded measurable function, $f$, defined on measurable set, $E$, with $\mu E < \infty$, define (Lebesgue) integral of $f$ over $E$ $$ \int_E f(x) dx = \sup_{\varphi:\ \mathrm{simple},\ \varphi \leq f} \int_E \varphi = \inf_{\psi:\ \mathrm{simple},\ f \leq \psi} \int_E \psi $$

Properties of Lebesgue integral of bounded functions

for bounded measurable functions, $f$ and $g$, defined on $E$ with finite measure
- for every $a,b\in\reals$ $$ \int_E (af+bg) = a \int_E f + b\int_E g $$
- if $f\leq g$ a.e. $$ \int_E f \leq \int_E g $$
- for disjoint measurable sets, $A,B\subset E$, $$ \int_{A\cup B} f = \int_A f + \int_B f $$
hence, $$ \left|\int_E f \right| \leq \int_E |f| \mbox{ \& } f=g \mbox{ a.e. } \Rightarrow \int_E f = \int_E g $$

Lebesgue integral of bounded functions over finite interval

if bounded function, $f$, defined on $[a,b]$ is Riemann integrable, then $f$ is measurable and $$ \int_{[a,b]} f = R \int_a^b f(x) dx $$ where $R\int$ denotes Riemann integral
bounded function, $f$, defined on $[a,b]$ is Riemann integrable if and only if set of points where $f$ is discontinuous has measure zero
for sequence of measurable functions, $\seq{f_n}$, defined on measurable $E$ with finite measure, and $M>0$, if $|f_n|<M$ for every $n$ and $f(x) = \lim f_n(x)$ for every $x\in E$ $$ \int_E f = \lim \int_E f_n $$

Lebesgue integral of nonnegative functions

for nonnegative measurable function, $f$, defined on measurable set, $E$, define $$ \int_E f = \sup_{h:\ \mathrm{bounded\ measurable\ function},\ \mu\set{x}{h(x)\neq0}<\infty,\ h\leq f} \int_E h $$
for nonnegative measurable functions, $f$ and $g$
- for every $a,b\geq0$ $$ \int_E (af + bg) = a\int_E f + b\int_E g $$
- if $f\geq g$ a.e. $$ \int_E f \leq \int_E g $$
thus,
- for every $c>0$ $$ \int_E cf = a\int_E f $$

Fatou's lemma and monotone convergence theorem for Lebesgue integral

Fatou's lemma - for nonnegative measurable function sequence, $\seq{f_n}$, with $\lim f_n = f$ a.e. on measurable set, $E$ $$ \int_E f \leq \liminf \int_E f_n $$
- note $\lim f_n$ is measurable (page~), hence $f$ is measurable (page~)
monotone convergence theorem - for nonnegative increasing measurable function sequence, $\seq{f_n}$, with $\lim f_n = f$ a.e. on measurable set, $E$ $$ \int_E f = \lim \int_E f_n $$
for nonnegative measure function, $f$, and sequence of disjoint measurable sets, $\seq{E_i}$, $$ \int_{\cup E_i} f = \sum \int_{E_i} f $$

Lebesgue integrability of nonnegative functions

nonnegative measurable function, $f$, said to be integrable over measurable set, $E$, if $$ \int_E f < \infty $$
for nonnegative measurable functions, $f$ and $g$, if $f$ is integrable on measurable set, $E$, and $g\leq f$ a.e. on $E$, then $g$ is integrable and $$ \int_E (f-g) = \int_E f - \int_E g $$
for nonnegative integrable function, $f$, defined on measurable set, $E$, and every $\epsilon$, exists $\delta >0$ such that for every measurable set $A\subset E$ with $\mu A< \epsilon$ (then $f$ is integrable on $A$, of course), $$ \int_A f < \epsilon $$

Lebesgue integral

for (any) function, $f$, define $f^+$ and $f^-$ such that for every $x$ $$ \begin{eqnarray*} f^+(x) &=& \max\{f(x), 0\} \\ f^-(x) &=& \max\{-f(x), 0\} \end{eqnarray*} $$
note $f = f^+ - f^-,\ |f| = f^+ + f^-,\ f^- = (-f)^+$
measurable function, $f$, said to be (Lebesgue) integrable over measurable set, $E$, if (nonnegative measurable) functions, $f^+$ and $f^-$, are integrable $$ \int_E f = \int_E f^+ - \int_E f^- $$

Properties of Lebesgue 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 $f\geq g$ a.e. on $E$, $$ \int_E f \geq \int_E g $$
- for disjoint measurable sets, $A,B\subset E$ $$ \int_{A\cup B} f = \int_A f + \int_B g $$

Lebesgue convergence theorem (for Lebesgue integral)

Lebesgue convergence theorem - for measurable $g$ integrable on measurable set, $E$, and measurable sequence $\seq{f_n}$ converging to $f$ with $|f_n|<g$ a.e. on $E$, ($f$ is measurable (page~), every $f_n$ is integrable (page~)) and $$ \int_E f = \lim \int_E f_n $$

Generalization of Lebesgue convergence theorem (for Lebesgue integral)

generalization of Lebesgue convergence theorem - for sequence of functions, $\seq{g_n}$, integrable on measurable set, $E$, converging to integrable $g$ a.e. on $E$, and sequence of measurable functions, $\seq{f_n}$, converging to $f$ a.e. on $E$ with $|f_n|<g_n$ a.e. on $E$, if $$ \int_E g = \lim \int_E g_n $$ then ($f$ is measurable (page~), every $f_n$ is integrable (page~)) and $$ \int_E f = \lim \int_E f_n $$

Comments on convergence theorems

Fatou's lemma (page~), monotone convergence theorem (page~), Lebesgue convergence theorem (page~), all state that under suitable conditions, we say something about $$ \int \lim f_n $$ in terms of $$ \lim \int f_n $$
Fatou's lemma requires weaker condition than Lebesgue convergence theorem, i.e., only requires “bounded below'' whereas Lebesgue converges theorem also requires “bounded above'' $$ \int \lim f_n \leq \liminf \int f_n $$
monotone convergence theorem is somewhat between the two;
- advantage - applicable even when $f$ not integrable
- Fatou's lemma and monotone converges theorem very clsoe in sense that can be derived from each other using only facts of positivity and linearity of integral

Convergence in measure

$\seq{f_n}$ of measurable functions said to converge $f$ in measure if $$ (\forall \epsilon>0) (\exists N\in\naturals) (\forall n > N) (\mu\set{x}{|f_n-f|>\epsilon} < \epsilon) $$
thus, third statement on page~ implies $$ (\forall \seq{f_n} \mbox{ converging to } f \mbox { a.e. on } E \mbox{ with } \mu E<\infty) (f_n \mbox{ converge in measure to }f) $$
however, the converse is not true, i.e., exists $\seq{f_n}$ converging in measure to $f$ that does not converge to $f$ a.e.
- e.g., XXX
Fatou's lemma (page~), monotone convergence theorem (page~), Lebesgue convergence theorem (page~) remain valid! even when “convergence a.e.'' replaced by “convergence in measure''

Conditions for convergence in measure

$$ \left( \forall \seq{f_n} \mbox{ converging in measure to } f \right) \left( \exists \mbox{ subsequence }\seq{f_{n_k}} \mbox{ converging a.e. to } f \right) $$

for sequence $\seq{f_n}$ measurable on $E$ with $\mu E<\infty$ $$ \begin{eqnarray*} &=&\seq{f_n} \mbox{ converging in measure to } f \\ &\Leftrightarrow& \left( \forall \mbox{ subsequence }\seq{f_{n_k}} \right) \left( \exists \mbox{ its subsequence }\seq{f_{n_{k_l}}} \mbox{ converging a.e. to } f \right) \end{eqnarray*} $$

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Sunghee Yun (He/Him)