Lebesgue measure (updating)

March 08, 2022

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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

$\phi, X \in \Sigma$,
$A\in\Sigma \implies A^c := X\backslash A \in\Sigma$, and
$\displaystyle\{A_i\}_{i\in\mathbb{N}} \implies \bigcup_i A_i \in \Sigma$. ($i$ 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

$\mu(\phi) = 0$
$\displaystyle\mu \left( \bigcap_{k=1}^{\infty} E_k \right) = \sum_{k=1}^{\infty} \mu(E_k)$, 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 $P$.

Vitali set

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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.

a

Reference

Partition of a set

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

$\{P_i\}$ is pairwise disjoint: $\forall P_i, P_j, P_i\cap P_j=\phi$ when $P_i\neq P_2$,
The union of $\{P_i\}$ forms the whole set $X$: $\displaystyle \bigcup_i P_i = X$
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.