Size

Measurable space

Suppose that $S$ is a $\sigma$ algebra over $X$. $(X,S)$ is called a measurable space. Every member of $S$ is a measurable set.

$(S,2^S)$ is a common measurable space.

Importance

This notion is useful because it enumerates the sets whose size we want to measure.

Product space

You can take two measurable spaces $(S_1,F_1),(S_2,F_2)$, and, by set product get a bigger measurable space $(S_1\times S2,F_1\times F_2)$.

Measure

Minimal definition

$m:S\to [0,\mathrm{\infty }]$, with $m(\varphi )=0$ and the countable additivity property ($A\cap B=\varphi ⟹m(A\cup B)=m(A)+m(B)$) is called a measure on $X$ of subsets $S$.

$(X,S,m)$ is called a measure space.

Motivation

This measure of size generalizes concepts such as volume/ area/ box measure, mass, time. Especially important measures are the box measure and the probability measure.

Special classes

General additivity

For some measures $m$, any bunch of mutually disjoint sets $S_i$: $m({\cup }_i{S}_i)=\sum _i{S}_i$. This is stronger than countable additivity.

Finite measures

If $X$ is a countable union of finite measure sets, $m$ is $\sigma -$finite. This is a very common property.

Signed measure

If $m(x)<0$ is allowed, then $m$ is a signed measure.

Null set, almost-everywhereness

If $m(T)=0$, then $T$ is called a $m-$null set.

A property (eg: $f(x)>0$) holds ‘almost everywhere’ if the set of elements for which the property does not hold is a null set. Eg: ‘Almost always’ in applications of probability.

Size of the union and intersection

Inclusion/ exclusion principle

The following holds for any measure which is finite on the sets involved. $|{\cup }_{i\in V}{S}_i|=\sum _i|S_i|-\sum _{i\ne j}|S_i\cap S_j|+..=\sum _{T\subseteq V}(-1{)}^{|T|+1}|{\cap }_{i\in T}{S}_i|$.

Bounds

Thence, we have the union upper bound: $m(A\cup B)\le m(A)+m(B)$. In the case of probabilistic analysis, this is very useful. Aka Boole’s inequality.

Intersection lower bound: $m(A\cap B)\ge m(A)+m(B)-1$. Aka Bonferroni’s inequality.

Proof

$m(A\cup B)\le 1$ with the inclusion-exclusion principle.

By mathematical induction, $m({\cap }_{i\in (1,n)}{A}_i)\ge \sum _{i}m(A_i)-(n-1)$.

Generalization

(Mobius inversion lemma). Got functions on sets f, g. $[\mathrm{\forall }A\subseteq V:f(A)=\sum _{B\subseteq A}g(B)]\equiv [g(B)=\sum _{B\subseteq A}(-1{)}^{A-B}f(B)]$. \chk Easy algebraic proof.

Counting measure

The counting measure of $A$, $|A|$, equals the number of elements in a set.

Counting, combinatorics

See probability survey.

Cardinality

The concept of Cardinal numbers extends the notion of the counting measure to compare the sizes of even infinite sets. For finite sets, the cardinality equals the counting measure.

Comparison by bijection

Comparison of cardinalities of A and B can be made by making bijections, even if they’re $\mathrm{\infty }$ sets: ‘Equinumerousness’.

Hierarchy of cardinal numbers

Consider the power set $P(S)$. $|P(S)|>|S|$.

Cardinalities of compared to N

Cardinality (or power) of the continuum $c=|R|$; ${\mathrm{\aleph }}_0=|N|$. Continuum hypothesis: $\mathrm{\exists }{c}^{\prime }?:{\mathrm{\aleph }}_0<c^{\prime }<c$.

Countability

$S$ is countable if it can be mapped to $N$.

Countable unions of countable $S$ still countable: write any $S$ as a row vector, see their sequence as a matrix, draw a zig-zag line to cover all matrix elements. Similarly, see countability of Q.

If $S$ is countable, $S^n$ is countable: use induction: if $S^{k-1}$ countable, for every $a\in S$, $\{ba:b\in S^{k-1}\}$ is countable; So, their union is also countable.

Show uncountability

Use Cantor’s diagonalization.

Infinite (sub)sets’ cardinality

(Dedekind): $S$ is $\mathrm{\infty }$ iff $\mathrm{\exists }A\left\{S\right\}$ with same cardinality as $S$.

Proof

Finite $S$ can’t have such a proper subset. If $|S|=\mathrm{\infty }$, get countably $\mathrm{\infty }$ $S^{\prime }$; map to $N$ with function $f$; but map $n\in N$ to $n+1$ with function g, do $f^{-1}$.

Product measure

Consider the product of two measure spaces: $\{(S_i,F_i,m_i)|i\in \{1,2\}\}$. The product measure: $m(E_1,E_2)=m_1(E_1)\times m_2(E_2):\mathrm{\forall }{E}_i\in S_i$.

By induction, one can define product measure for the product of arbitrary number of measure spaces.

Importance

This measure finds application in defining, for example, measures for $R^n$ based on the common box measure for $R$; and in considering measures over the product of ranges of multiple random variables.

Extension to bigger sigma algebra

Consider the product space with an expanded sigma algebra $T$ such that $(F_1\times F_2)\subseteq T\subseteq 2^{S_1}\times 2^{S_2}$.

For $A\in T$, one can use the product measure $m$ to define the minimum cover measure $m^{\prime }(A)=inf\{\sum _{i}m(B_i):A\subseteq {\cup }_i{B}_i\}$. One can show that this obeys required properties like countable additivity.

This is aka Lebesgue measure.

Importance

It forms a natural basis for defining and studying the box integral over product of multiple measure spaces.

Connecting measures

Absolute continuity

Consider two $\sigma -$finite measures $m,n$. $m$ is absolutely continuous with $n$ - or $m$ is dominated by $n$ - or $m«n$ if $\mathrm{\forall }t\in S:n(t)=0⟹m(t)=0$.

This is equivalent to a definition reminiscent of absolute continuity of functions $n,m$: $\mathrm{\forall }ϵ,\mathrm{\exists }\delta :\mathrm{\forall }t:n(t)\le \delta ⟹m(t)\le ϵ$. Prior definition implies this because if there $\mathrm{\exists }ϵ,t:m(t)\ge ϵ\wedge (n(t)\le \delta \mathrm{\forall }\delta )$ then for that t, $m(t)\ge ϵ$ while $n(t)=0$.

This notion is important in defining the inter-measure derivative.

Inter-measure Derivative

Aka Radon-Nikodym derivative. For measures $m«n$ over $(X,S)$, a theorem by Radon/ Nikodym says that $\mathrm{\exists }f:X\to [0,\mathrm{\infty }]:m(t)={\int }_{x\in t}f(x)dn$, and that this $f$ is unique almost everywhere wrt $n$. \why

Note that $m«n$ is necessary: otherwise, for the event $E$ where $n(E)=0,m(E)\ne 0$, there is no $f$ such that: $m(E)={\int }_{E}f(x)dn$.

This concept is important in defining probability density functions of random variables.