Lebesgue Outer Measure

by | Nov 21, 2021 | Math Learning

Introduction:

The Riemann integral of a bounded function over a closed, bounded interval is defined using approximations of the function that are associated with partitions of its domain into finite collections of subintervals. The generalization of the Riemann integral to the Lebesgue integral will be achieved by using approximations of the function that are associated with decompositions of its domain into finite collections of sets which we call Lebesgue measurable. Each interval is Lebesgue measurable. The richness of the collection of Lebesgue measurable sets provides better upper and lower approximations of a function, and therefore of its integral, than are possible by just employing intervals. This leads to a larger class of functions that are Lebesgue integrable over very general domains and an integral that has better properties. For instance, under quite general circumstances we will prove that if a sequence of functions converges point-wise to a limiting function, then the integral of the limit function is the limit of the integrals of the approximating functions.

Motivation:

One of the primary motivations for developing a theory has to do with the failure of the theory of Riemann integration to behave nicely undertaking point-wise limits of functions. In particular, it is well-known that point-wise convergence of a sequence of functions does not translate to the interchange of limits and integrals; for example, if

    \[f_{n}(x)=\begin{cases} 1, \;\;\; & \text{when } x\in [n,n+1] \\ 0, \;\;\; & \text{otherwise } \end{cases}\]

then as n\to \inftyf_{n} converges point wise to the zero function. If we take its integral,

                                     

    \[\int_{-\infty}^{\infty}f_{n}(x)dx=1\]

Which leads to

    \[\lim_{n\to\infty}\int_{-\infty}^{\infty}f_{n}(x)dx=1\]

But

    \[\int_{-\infty}^{\infty}\lim_{n\to\infty}f_{n}(x)dx=0\]

This inability to interchange a limit and an integral can be extremely inconvenient. One can interchange a limit and Riemann integration under certain circumstances like uniform convergence, but this is usually far too restrictive of a condition. What measure-theoretic integration gives you is a way to do this interchange provided us, have point-wise convergence and much weaker conditions than uniform convergence; in exchange, it forces us to throw out sets of measure zero. In addition, once the theory is developed we suddenly have extremely powerful tools to take limits of integrals you did not have before (Lebesgue dominated convergence theorem).

So we have two main motivations to make a rich theory

  1. To integrate more functions (Like Dirichlet Function )
  2. To make a space this is completely integrable.

There are more motivations like

  1. Interchanging limits and integrals (e.g. the limit of a sequence of continuous functions may not be Riemann integrable)
  2. Length of Curves (finding the length of curves that are only rectifiable, not continuously differentiable), etc.

Definition:

To define Lebesgue Measure on a set we need some precious definitions and some important functions and classes or collection of subsets of \mathbb{R}.

Length Function:

Let \mathbb{g} denote the collection of all intervals of \mathbb{R}. If an interval I\in\mathbb{g} has
end points a and b we write it as I (a, b). By convention, the open interval
(a,a)=0 \;\; \forall a\in\mathbb{R}. . Let [0,+\infty[=\lbrace x\in\mathbb{R}^{*}\,:\, x\geq 0\rbrace=[0,+\infty)\cup\lbrace\infty\rbrace.
Define the function \lambda:\; \mathbb{g}\to\mathbb{R}^{*} by

    \[\lambda((a,b))=\begin{cases} \vert b-a\vert, \;\;\; & \text{if } a,b\in \mathbb{R} \\ +\infty, \;\;\; & \text{if either } a=-\infty \text{ or } b=+\infty \text{ or both} \end{cases}\]

This \lambda is called length function defined on \mathbb{g}.

 

Properties of Set Function:

  1. \lambda(\emptyset)=0
  2. \lambda is monotonic i.e. \lambda(I)\leq\lambda(J) \text{ when } I\subseteq J
  3. \lambda is finitely additive.

Let I\in \mathbb{g} be such that \displaystyle I=\cup_{i=1}^{n}J_{i} where J_{m}\cap J_{n}\neq\emptyset for m\neq n Then

    \[\lambda(I)=\sum_{i=1}^{n}\lambda(J_{i}) \]

  1. \lambda is finitely sub-additive.

Let I\in\mathcal{I} be such that \displaystyle I\subseteq \bigcup_{i=1}^{n}J_{i}  Then

    \[\lambda(I)\leq \sum_{i=1}^{n}\lambda(J_{i}) \]

  1. \lambda is countable additive .

Let I\in \mathcal{I}  be such that \displaystyle I= \bigcup_{i=1}^{\infty}J_{i} where J_{m}\cap J_{n}\neq \emptyset for m\neq n Then

    \[\lambda(I)= \sum_{i=1}^{\infty}\lambda(J_{i}) \]

  1. \lambda is countable sub-additive .

Let I\in \mathcal{I} be such that I\subseteq \bigcup_{i=1}^{\infty}J_{i} where

Then

    \[\lambda(I)\leq \sum_{i=1}^{\infty}\lambda(J_{i}) \]

  1. Translation Invariance

    \[\lambda(I)=\lambda(I+x),\]

For every I\in\mathbb{g} and x\in\mathbb{R}.

 

Algebra on a set:

Let X be any non-empty set and let \mathcal{C} a collection of subsets of X. Then the collection \mathcal{C} is called an algebra of subsets of X if  \mathcal{C} satisfies following properties:

  1. \emptyset, X \in\mathcal{C}
  2. A\cap B\in \mathcal{C} whenever A,B\in\mathcal{C}
  3. A^{c}\in\mathcal{C} whenever A\in\mathcal{C}

Then \mathcal{C} is called Algebra on X.

 

\sigma- Algebra on a set:

Let X be any non-empty set and let \mathcal{S} a collection of subsets of X. Then the collection \mathcal{S} is called an \sigma-Algebra of subsets of X if  \mathcal{S} satisfies following properties:-

  1. \emptyset, X \in\mathcal{S}
  2. A^{c}\in\mathcal{S} whenever A\in\mathcal{S}
  3. \cup_{i=1}^{n} A_{i}\in\mathcal{S} for A_{i}\in\mathcal{S}\;\;\; i=1,2,3,\cdots\cdots\cdots

Then \mathcal{S} is called \sigma-Algebra on X.

 

Definition of Lebesgue Measure:

Def:- For every subsets A of \mathbb{R}, the Lebesgue Outer Measure of A, denoted by \mathit{m}^{*}(A), is defined by

    \[\displaystyle\mathit{m}^{*}(A)=\inf\left\lbrace\sum_{i=1}^{\infty}\lambda(I_{i}):A\subset\bigcup_{i=1}^{n}I_{i}\right\rbrace\]

Where \lbrace I_{n}\rbrace varies over all possible sequence of open intervals of \mathbb{R} whose union contains A.

Properties of Lebesgue Outer Measure:

The Lebesgue Outer Measure is generated by length function which is defined on earlier so it’s preserves some of their properties.

  1. \mathit{m}^{*}(\emptyset)=0
  2. \mathit{m}<u>^{*}</u> is monotonic i.e. \mathit{m}^{*}(I)\leq\mathit{m}^{*}(J) \text{ when } I\subseteq J
  3. \mathit{m}^{*} is finitely sub-additive .

Let I\in\mathcal{I} be such that I\subseteq\bigcup_{i=1}^{n}J_{i} Then

    \[\mathit{m}^{*}(I)\leq \sum_{i=1}^{n}\mathit{m}^{*}(J_{i}) \]

  1. \mathit{m}^{*} is countable sub-additive .

Let I\in\mathcal{I} be such that I\subseteq\bigcup_{i=1}^{\infty}J_{i} where

Then

    \[\mathit{m}^{*}(I)\leq \sum_{i=1}^{\infty}\mathit{m}^{*}(J_{i}) \]

  1. Translation Invariance

    \[\mathit{m}^{*}(I)=\mathit{m}^{*}(I+x),\]

For every I\in\mathbb{g} and x\in\mathbb{R}.

Now we already noticed that \mathit{m}^{*} hasn’t had the properties of finitely additive and not countable additive (why?)

A Vitali* Set in [0,1], has a positive measure

Specifically, let \oplus denote translation. (\textit{That is, for A\subset [0,1] and x\in [0,1] let A \oplus x =\lbrace a+x, a+x-1:a\in A\rbrace\cap[0,1]}.) Note that outer measure \mathit{m}^{*} is invariant under translation, so

    \[\mathit{m}^{*}(A\oplus x)=\mathit{m}^{*}(A)\]

Now let V be a Vitali set, and let q_{1},q_{2},\ldots be an enumeration of the rationals in [0,1]. By construction of V, the sets V\oplus q_{1}, V\oplus q_{2}, \ldots are pair wise disjoints and their union is [0, 1]. By countable sub-additively we have

    \[1=\mathit{m}^{*}([0,1])\leq\sum_{n}\mathit{m}^{*}(V\oplus q_{n})=\sum_{n}\mathit{m}^{*}(V)=\mathit{m}^{*}([0,1])\]

In particular we must have \mathit{m}^{*}(V)>0

So we can find an integer \mathbb{N} sufficiently large that

    \[N.\mathit{m}^{*}(V)>1, \text{ For } n=1,\ldots,N\]

,

Let E_{n}=V\oplus q_{n}.

Then the sets E_{n} are pair wise disjoint, and since E_{n}\subset[0,1]

We have

    \[E=\bigcup_{n=1}^{N}E_{n}\]

.

Hence by monotonicity, \mathit{m}^{*}(E)\leq \mathit{m}^{*}([0,1])=1.

On the other hand,

    \[\sum_{n=1}^{N}\mathit{m}^{*}(E_{n})=\sum_{n=1}^{N}\mathit{m}^{*}(V\oplus q_{n})=\sum_{n=1}^{N}\mathit{m}^{*}(V)=N.\mathit{m}^{*}(V)>1.\]

So we have

    \[\mathit{m}^{*}(E)<\sum_{n=1}^{N}\mathit{m}^{*}(E_{n})\]

So we saw that \mathit{m}^{*} is not finitely additive and it is also not countable sub-additive.

Vitali: An elementary example of a set of real numbers which is not Lebesgue measurable.

Definition:

Measurable Set(Carathéodory Condition): A set E\subset\mathbb{R} is said to be Lebesgue-Measurable if

    \[\mathit{m}^{*}(A)=\mathit{m}^{*}(A\cap E)+\mathit{m}^{*}(A\cap E^{c}) \text{ for all } A\subset \mathbb{R}\]

This Condition is known as Carathéodory Condition.

 

Properties of Measurable set in R

  1. If E is Lebesgue Measurable then E^{c} also Lebesgue Measurable set in \mathbb{R}
  2. \mathbb{R} and \emptyset are Lebesgue Measurable
  3. Every Interval in \mathbb{R} is Lebesgue Measurable.
  4. Every countable set is Lebesgue Measurable with measure zero i.e.

If B is a countable set in \mathbb{R} then

    \[\mathit{m}^{*}(B)=0.\]

  1. Every uncountable set has a non-zero measure (except Cantor Set)
  2. Every Borel* set is Lebesgue Measurable.

(Borel Set: Collection of all F_{\sigma} and G_{\delta} sets)

Definition:

Class \mathcal{M}:- Collection of all Lebesgue Measurable sets in \mathbb{R}.

    \[\mathcal{M}=\lbrace E\; : \; \text{E is Lebesgue meusurable in }\mathbb{R}; E\subseteq\mathbb{R} \rbrace\]

This Collection \mathcal{M} is a \sigma-Algebra and it is the largest measurable \sigma-Algebra over \mathbb{R} and (\mathbb{R},\mathcal{M},\mathit{m}^{*})  is called Measure Space which we want to achive.

Conclusion:

Now we got a new developed theory on subsets of \mathbb{R} and we can now use the theory to integrate those kinds of functions which we can’t integrate by using Riemann Integration. Also, we can use the theory to find the length of those kinds of curves which is not continuously differentiable on \mathbb{R} and we can use to find the integration over rationalize functions too. This rich theory gave us more freedom to apply on the space or subsets in \mathbb{R}.

Bibliography: 

  1. Yeh ; Real Analysis
  2. Robert G. Bartle; The Elements of Integration and Lebesgue Measure.
  3. Sheldon Axler; Measure, Integration, and Real Analysis.

 Thank You.

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