Introduction to Measure Theory

--- tags: analysis, measure theory --- # Introduction to Measure Theory ## Table of Contents [TOC] ## Introduction ### Sigma Algebra and Borel Sets **Def** $X$: a set, $\mathscr A \subseteq \mathcal P (X)$. $\mathscr A$ is a $\sigma$-algebra on $X$ if (1) $X \in \mathscr A$ (2) $\forall A \in \mathscr A, A^C \in \mathscr A$ (3) $\forall A_n \in \mathscr A (n \in \mathbb N),\cup_{n = 1}^{\infty} A_n \in \mathscr A$ Remark 結合(2)和(3)的結論，可以得到$\cap_{n = 1}^{\infty} A_n \in \mathscr A$. Intuition: 給定$X$，我們希望在$X$的某些子集上定義體積or面積，並且這些子集可形成一個$\sigma$-algebra。（如此一來，會帶來許多良好性質） **Def** A ***measurable space*** $X = (\underline X, m_X)$ consists of a set $\underline X$ and a $\sigma$-algebra on $\underline X$. And we say that $A$ is $m_X$-measurable if $A \in m_X$. **Def** $(X, m_x), (Y, m_y)$: measurable space. A ***measurable map*** $f$ (or we say that $f$ is $(m_X, m_Y)$-measurable) is a map $X \overset f \rightarrow Y$ such that $$ \forall B \in m_Y, f^{-1}(B) \in m_x. $$ Intuition: 可類比於拓墣空間中連續函數的性質(此類映射能夠保存measureable space的某些良好性質)。 Example Let $X$ be a set and $\mathscr A_j (j \in J)$ is a family of $\sigma$-algebra on $X$. Then it can be easily shown that $\cap_{j \in J} \mathscr A_j$ is also a $\sigma$-algebra on $X$. In particular, for any $\mathscr S \subseteq \mathcal P(X)$, there exists a smallest $\sigma$-algebra containing $\mathscr S$: $$ \mathscr A (S) := \cap \{ \mathscr A | \mathscr A: \sigma\text{-algebra containing } S \}. $$ $\mathscr A (S)$ is called the $\sigma$-algebra generated by $S$. Unfortunely, there is no general way to explicitly describe $\mathscr A (S)$. Example (1) Let $X = (\underline X, \mathscr O_X)$ be a topology space. The set $$ \mathscr B_X := \mathscr A (\mathscr O_x) $$ is called the ***Borel sets*** of $X$. 由$\sigma$-algebra的定義可知，該拓墣下所有可數多開集之交集(denoted by $G_\delta$)和可數多閉集之交集(denoted by $F_\delta$)皆包含於Borel sets之中。 ### Positive Measure and Measurable Space **Def** Let $(X, m)$ be a measurable space. A function $\mu: m \rightarrow [0, \infty] = [0, \infty) \cup \{\infty\} = \overline {\mathbb R}_+$ is a positive measure if (1) $A_n \in m, n \in \mathbb N ,A_i \cap A_j = \phi \ \forall i, j$. Then we have $$ \mu ( \cup_{n = 1}^ \infty A_n ) = \sum_{n=1}^\infty \mu (A_n) $$ We say that $\mu$ is $\sigma$-additive or countably additive. (2) $\exists A \in m, \mu(A) < \infty$. (排除一些無聊的measure) Remark 在(1)成立的前提下，(2)等價於$\mu (\phi) = 0$ *proof* Suppose $\mu(\phi) \neq 0$. For any $A \in m$, $\mu(A) \overset{(1)} = \mu(A) + U_{i = 0}^\infty \mu (\phi) = \infty$. (a contradiction) **Def** (1) A measure space $(X, m , \mu)$ consists of a measurable space $(X, m)$ and a positive measure $\mu$ on $(X, m)$. (measurable space只定義了哪些集合是適合測量容積的，卻未定義測量的方式) (2) If $\mu(X) = 1$, we say that $(X, m, \mu)$ is a probability space. Note For $\overline {\mathbb R} := \{ -\infty, \infty \} \cup \mathbb R$ (actually, we can regard $\overline {\mathbb R}$ as the union of real numbers and a set disjoint to $\mathbb R$ with whatever two elements), we topologize $\overline {\mathbb R}$ such that the map $\overline {\mathbb R} \rightarrow [ - \frac{\pi}{2}, \frac{\pi}{2} ]$ is a homeomorphism $f$ where $$ f = \begin{cases} \tan ^{-1} x, & \text{if } x \in \mathbb R \\ \frac{\pi}{2}, & \text{if } x = \infty \\ -\frac{\pi}{2}, & \text{if } x = -\infty \\ \end{cases} $$ $\overline{\mathbb R}_+ := [0, \infty) \cup \{ \infty \}$ is equipped with the subspace topology. For addition in $\overline {\mathbb R}$, $x + y$ is defined except in the cases where $x = \infty, y = - \infty$ (and vice versa): $$ x + y = \begin{cases} x + y \text { in real field}, & \text{if } x, y \in \mathbb R \\ \infty, & \text{if } x \in \mathbb R, y = \infty \text{ (and vice versa)} \\ -\infty, & \text{if } x \in \mathbb R, y = -\infty \text{ (and vice versa)} \end{cases} $$ For mutiplication, we define $$ x \times \infty = \begin{cases} \infty, & \text{if } x \in (0, \infty) \\ -\infty, & \text{if } x \in (-\infty, 0) \\ 0, & \text{if } x = 0 \end{cases} $$ The definition for $- \infty$ is similar. **Def** A map $X \overset{f}\rightarrow Y$ between topology spaces is Borel measurable if it is $(\mathscr B_X, \mathscr B_Y)$-measurable. Example (1) $(X,\mathscr m)$: a measurable space $X \overset{f}{\rightarrow} Y$: a map to a set $Y$ If $\mathcal S \subseteq \mathcal P(Y)$, then $$ (X, \mathcal m) \overset{f}{\rightarrow} (Y, \mathscr A(\mathcal S)) \text{ is measurable} \iff \forall S \in \mathcal S, f^{-1}(S) \in m $$ In other words, to check the measurability of $f$, we only need to consider the elements of $\mathcal S$. proof. ($\Rightarrow$) Trivial result by definition. ($\Leftarrow$) Suppose that $\forall S \in \mathcal S, f^{-1}(S) \in m$. Consider the set $\mathcal E = \{E | E \in \mathcal P(Y), f^{-1}(E) \in \mathcal m\}$. It can be easily shown that $\mathcal E$ is also a sigma algebra. On the other hand, by the definition of $\mathcal E$, $\mathcal S \subseteq \mathcal E$ and thus $\mathscr A(\mathcal S) \subseteq \mathcal E$ (by the definition of $\mathscr A(\mathcal S)$). Thus, for each element in $\mathscr A(\mathcal S)$, its pre-image is also in $\mathcal m$, which completes the proof. (2) $X \overset{f}{\rightarrow} Y$: a continuous map (between topology spaces) Then $f$ is Borel measurable. This is a direct result from (1). (3) If $X$ is a topology space and $Y \subseteq X$ is a subspace, then $\mathscr B_Y = \{ \mathcal B \cap Y | \mathcal{B} \in \mathscr B_X \}$ Example (1) (2) For any two topology spaces $X$ and $Y$, we have $\mathscr B_X \times \mathscr B_Y \subseteq \mathscr B_{X \times Y}$. Furthermore, if $X$ and $Y$ are both second countable (e.g. $\mathbb R^n$), then $\mathscr B_X \times \mathscr B_Y = \mathscr B_{X \times Y}$. ## Basic Properties of Measurable Spaces Proposition $(X, \mathscr m, \mu)$: measure space. (1) Monotonicity $\forall A, B \in \mathscr m$, if $A \subseteq B$, then $\mu(A) \le \mu(B)$. Furthermore, if $\mu(A) < \infty$, then $\mu(B \setminus A) = \mu(B) - \mu(A)$. (2) Substractabiliy $$ \forall A_n \in m (n \in \mathbb N), \mu(\bigcup_{n=1}^\infty A_n) \le \sum_{n=1}^\infty \mu(A_n) $$ (3) For any sequence of measurable sets $A_n \in \mathscr m (n \in \mathbb N)$, if $A_n \uparrow A$ (must in $m$) (monotone), then $$ \lim_{n \rightarrow \infty} \mu(A_n) = \mu(A) $$ Similarly, if $A_n \downarrow A$ **and** $\exists m \in \mathbb N$ such that $\mu(A_m) < \infty$, then $$ \lim_{n \rightarrow} \mu(A_n) = \mu(A) $$ proof. (1) We have $$ B \setminus A = B \cap A^C \in n $$ and $$ (B \setminus A) \cap A = \phi $$ Thus, $$ \mu(B) = \mu(A) + \mu(B \setminus A) \ge \mu(A) $$ It can be checked that $\mu(B) - \mu(A) = \mu(B \setminus A)$ (2) Let $$ C_m := A_m \setminus (A_1 \cup \cdots \cup A_m) $$ and $$ A_0 := \phi $$ Apparently, $C_m(m \in \mathbb N)$ is disjoint and $A_1 \cup \cdots \cup A_m = C_1 \cup \cdots \cup C_m, \forall m \in \mathbb N$. Thus, $$ \mu \left(\bigcup_{n=1}^{\infty} A_n\right) = \mu \left( \bigcup_{n=1}^{\infty} C_n\right) = \sum_{n=1}^{\infty} \mu(C_n) \le \sum_{n=1}^{\infty} \mu(A_n) $$ (3) Let $D_m = A_m \setminus A_{m-1} \in \mathscr m, m \in \mathbb N$ and $A_0 = \phi$ (Hence $D_m$ is disjoint). We have $$ A = \bigcup_{n=1}^{\infty} A_n = \bigcup_{n=1}^{\infty} D_n $$ Therefore, $$ \begin{align*} \mu(A) &= \sum_{n=1}^{\infty} \mu(D_n) \\ &= \lim_{m \rightarrow \infty} \sum_{n=1}^{m} \mu(D_n) \\ &= \lim_{m \rightarrow \infty} \mu(D_1 \cup \cdots \cup D_m) \\ &= \lim_{m \rightarrow \infty} \mu(A_m) \\ \end{align*} $$ Similarly, suppose $A_n \downarrow A$ **and** $\exists m \in \mathbb N$ such that $\mu(A_m) < \infty$. Observe that $A_k (k \in \mathbb N)$, can be expressed as union of two disjoint sets: $$ A_{k} = \left( \bigcap_{n=1}^{\infty} A_n \right) \cup \left( \bigcup_{n=k}^{\infty} A_n \setminus A_{n+1} \right) $$ This can be illustrated as the following figure: ![](https://i.imgur.com/MquBfQi.png =400x) Therefore, we have $$ \begin{align} \mu(A_k) &= \mu\left( \bigcap_{n=1}^{\infty} A_n \right) +\mu\left( \bigcup_{n=k}^{\infty} A_n \setminus A_{n+1} \right) \\ &= \mu\left( \bigcap_{n=1}^{\infty} A_n \right) +\sum_{n=k}^{\infty} \mu\left( A_n \setminus A_{n+1} \right) \\ &= \mu\left( \bigcap_{n=1}^{\infty} A_n \right) +\sum_{n=k}^{\infty} \left(\mu(A_n) - \mu(A_{n+1}) \right) \\ \end{align} $$ ### Examples of Measures 1. Point Mass Let $X$ be a set and $p \in X$. Define $\delta_p: \mathcal P(X) \rightarrow \overline{\mathbb R}_+ = [0, \infty]$ as follows: $$ E \overset{\delta_p}\mapsto \begin{cases} 1 & \text{ if } p \in E \\ 0 & \text{ if } p \notin E \end{cases} $$ Then $\delta_p$ is a positive measure. It's also called ***Dirac measure***. 2. Counting Measure $X$: a set. $$ E \mapsto \begin{cases} |E| & \text{ if } |E| < \infty \\ \infty & \text{ if } |E| = \infty \end{cases} $$ We can use counting measure to construct a sequence of sets $\{A_n\}$ such that $A_n \downarrow A$ but $\lim_{n\rightarrow \infty} \mu(A_n) \neq \mu(A)$: Let $X$ be $\mathbb N$ and $A_n = \{k \in X| k >= n\}$. In this way, $A = \phi$ but $\lim_{n \rightarrow \infty} \mu(A_n) = \infty$.