# Lebesgue measure (updating)

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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.

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

1. $\phi, X \in \Sigma$,
2. $A\in\Sigma \implies A^c := X\backslash A \in\Sigma$, and
3. $\displaystyle\{A_i\}_{i\in\mathbb{N}} \implies \bigcup_i A_i \in \Sigma$. ($i$ could be countable infinite).

The $\Sigma$ 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 $\displaystyle\bigcap_i \Sigma_i$ 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

$$\mathcal{B}(X) := \sigma(\tau)$$

## Measure space

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

A map $\mu:\Sigma\rightarrow [0,\infty]$ is caleed measure if

1. $\mu(\phi) = 0$
2. $\displaystyle\mu \left( \bigcap_{k=1}^{\infty} E_k \right) = \sum_{k=1}^{\infty} \mu(E_k)$, where $\displaystyle\left\{E_k\right\}_{k=1}^{\infty}$ is any countable collection of pairwise disjoint sets ($E_i\cap E_j = \phi$ if $i\neq j$ for all $i, j$)

$\left(X, \Sigma, \mu\right)$ 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

$$\mu(A) = \begin{cases} |A|, & \text{if } A \text{ is finite} \\\ \infty, & \text{if } A \text{ is infinite} \end{cases}$$

### Dirac measure

For $p\in X$,

$$\delta_{p}(\Sigma)= \begin{cases} 1, & \text{if } p\in\Sigma \\\ 0, & \text{else} \end{cases}$$

## Almost everywhere

“Almost everywhere” is a mathmatical terminology.

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

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## Measurable function (measurable map)

Let $(X_1,\Sigma_1)$ and $(X_2, \Sigma_2)$ be measurable spaces.

A function $f:X_1\rightarrow X_2$ is called measurable map if $f^{-1}(A_2)\in \Sigma_1$ for all $A_2\in\Sigma_2$.

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 $\mathbf{1}_{A}\colon X\to \{0,1\}$, defined as

$$\mathbf{1}_{A}(x):= \begin{cases} 1, &{\text{ if }}~x\in A,\\\ 0, &{\text{ if }}~x\notin A. \end{cases}$$

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 $(X, \Sigma)$ 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:

$$f(X) = \left\{f_1, f_2, \cdots , f_N | f_i\in\mathbb{R}\right\} \text{ for some } N\in\mathbb{N}$$

Note that, by definition of measurable functions, $\displaystyle f^{-1}\left(\left\{f_i\right\}\right) \in \Sigma$ 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 $\phi$.

The simple functions are closed under addition and multiplication.

### Representation of a simple function

We can write the equivalent simple function as follows:

$$f = \sum_{f_i\in f(X)} f_i \cdot \mathbf{1}_{f^{-1}\left(\left\{f_i\right\}\right)}$$

For example:

$$f(a_1) = \sum_{f_i\in f(X)} f_i \cdot \left[ \mathbf{1}_{f^{-1}\left(\left\{f_i\right\}\right)} \left(a_1\right) \right]$$

## 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:

$$\int_{X} f d\mu := \mathrm{sup} \left\{\sum_{h_i\in h(X)} z\cdot \mu\left(h^{-1}\left(\{h_i\}\right)\right) \ \bigg|\ h \text{ is a simple function, and } 0\leq h\leq f\right\}$$

• $f$ 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, $h^{-1}\left(\{h_i\}\right) \neq h^{-1}\left(\{h_j\}\right)$ when $i\neq j$ (pairwise disjoint).
• This integral is defined with supremum.

## Reference

### 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: $\forall P_i, P_j, P_i\cap P_j=\phi$ when $P_i\neq P_2$,
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: $\forall P_i, P_i\neq\phi$.

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