Lebesgue measure (updating)

My personal motivation(s):

• want to know when it is allowed to swap limit and sum.
• want to know when it is allowed to swap limit and integral.
• want to know what the Lebesgue integral is.
• want to make a base for probability theory.

$\sigma$-algebra

Suppose a set $X$ and its power set $\mathcal{P}(X)$.

The “measure” will be defined for some elements of $\mathcal{P}(X)$ later.

$\mathrm{\Sigma }\subseteq \mathcal{P}(X)$ is $\sigma$-algebra if

1. $\varphi ,X\in \mathrm{\Sigma }$,
2. $A\in \mathrm{\Sigma }⟹A^c:=X\mathrm{\setminus }A\in \mathrm{\Sigma }$, and
3. ${\displaystyle \{A_i{\}}_{i\in \mathbb{N}}⟹\bigcup _i{A}_i\in \mathrm{\Sigma }}$. ($i$ could be countable infinite).

The $\mathrm{\Sigma }$ is called ($A$-)measurable set.

The condition 2 is a difference from the definitioin of a tpological space.

Note that, we could have more than one $\sigma$-algebra for a set, depends on how to choose the subset of its power set.

Theorem: conjunction of $\sigma$-algebras

Arbitrary numbers of conjunction of sigma algebra ${\displaystyle \bigcap _i{\mathrm{\Sigma }}_i}$ is also a $\sigma$-algebra.

Generated $\sigma$-algebra

Let $F\in \mathcal{P}(X)$. There exists a unique smallest $\sigma$-algebra which contains every set in $F$. This $\sigma$-algebra is called “$\sigma$-algebra benerated by the family $F$”, denoted by $\sigma (F)$.

Borel ($\sigma$-)algebra

Let $(X,\tau )$ be a topological space. A Borel $\sigma$-algebra of $X$, $\mathcal{B}(X)$, is defined by

Measure space

Let $(X,\mathrm{\Sigma })$ be an measurable set.

A map $\mu :\mathrm{\Sigma }\to [0,\mathrm{\infty }]$ is caleed measure if

1. $\mu (\varphi )=0$
2. ${\displaystyle \mu \left(\bigcap _{k=1}^{\mathrm{\infty }}{E}_k\right)=\sum _{k=1}^{\mathrm{\infty }}\mu (E_k)}$, where ${\displaystyle {\left\{E_k\right\}}_{k=1}^{\mathrm{\infty }}}$ is any countable collection of pairwise disjoint sets ($E_i\cap E_j=\varphi$ if $i\ne j$ for all $i,j$)

$\left(X,\mathrm{\Sigma },\mu \right)$ is called measure space (cf. measurable space).

The second property is called $\sigma$-additivity.

Note that,

• In general, two elements of $\sigma$-algebra can have joint elements.

Examples of measure

Counting measure

Dirac measure

For $p\in X$,

Almost everywhere

“Almost everywhere” is a mathmatical terminology.

If $(X,\mathrm{\Sigma },\mu )$ is a measure space, a property $P$ is said to hold almost everywhere in $X$ if there exists a set $N\in \mathrm{\Sigma }$ with $\mu (N)=0$, and $\mathrm{\forall }x\in X\setminus N$ have the property $P$.

Vitali set

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Measurable function (measurable map)

Let $(X_1,{\mathrm{\Sigma }}_1)$ and $(X_2,{\mathrm{\Sigma }}_2)$ be measurable spaces.

A function $f:X_1\to X_2$ is called measurable map if $f^{-1}(A_2)\in {\mathrm{\Sigma }}_1$ for all $A_2\in {\mathrm{\Sigma }}_2$.

This map “preserve” the structure of measure spaces.

Properties

• When $f$ and $g$ is measurable functions, then $g\cdot f$ also.
• $f\pm g$ also.

Indicator function

The indicator function of a subset $A$ of a set $X$ is a function ${\mathbf{1}}_A:X\to \{0,1\}$, defined as

If $A$ contains $x$, it returns true (1), otherwise false (0).

In Physics, it is frequently called “characteristic function” and denote ${\chi }_A$.

Simple function and its representation

Simple function

Let $(X,\mathrm{\Sigma })$ and $\left(\mathbb{R},\mathcal{B}(\mathbb{R})\right)$ be a measurable spaces. The topology is the standard topology on $\mathbb{R}$.

A measurable function $f:X\to \mathbb{R}$ is called a simple function if the image of the entire set has the finite number of elements:

Note that, by definition of measurable functions, ${\displaystyle f^{-1}\left(\left\{f_i\right\}\right)\in \mathrm{\Sigma }}$ where $\{f_i\}\in \mathcal{B}(\mathbb{R})$.

When you imagine $X$ as a simple finite set, most pre-images in $\mathcal{B}(\mathbb{R})$ are $\varphi$.

The simple functions are closed under addition and multiplication.

Representation of a simple function

We can write the equivalent simple function as follows:

For example:

Integral of a measurable function

When the measurable space is induced a measure $\mu$, which always returns finite values, the integral of a measurable function $f$ with respect to $\mu$ is defined as follows:

• $f$ and $h$ take the same set $X$ which has the same $\sigma$-algebra.
• The differences between $f$ and $h$ is, $f$ is just a measurable function where $h$ is a simple function.
• By definition, $h^{-1}\left(\{h_i\}\right)\ne h^{-1}\left(\{h_j\}\right)$ when $i\ne j$ (pairwise disjoint).
• This integral is defined with supremum.

a

Reference

• https://www.youtube.com/watch?v=xZ69KEg7ccU&list=PLBh2i93oe2qvMVqAzsX1Kuv6-4fjazZ8j&index=1
• https://www.youtube.com/watch?v=ot253Lhx2_o
• https://web.ma.utexas.edu/users/gordanz/notes/lebesgue_integration.pdf

Partition of a set

A partition of a set $X$ is a set of subsets $\{P_i\}$ of $X$ such that:

1. $\{P_i\}$ is pairwise disjoint: $\mathrm{\forall }{P}_i,P_j,P_i\cap P_j=\varphi$ when $P_i\ne P_2$,
2. The union of $\{P_i\}$ forms the whole set $X$: ${\displaystyle \bigcup _i{P}_i=X}$
3. None of the elements of S is empty: $\mathrm{\forall }{P}_i,P_i\ne \varphi$.

When the number of partition is finite, the partition is called finite expansion.