Measure Theory

--- title: Measure Theory tags: cs 593 rl ---  ### Topological Spaces Let $X$ be a set with a collection $G$ of subsets of $X$. $G$ is a **topology** on $X$ iff: (1) $\varnothing, X \in G$ (2) G is closed under arbitrary union: For any subcollection $\{U_\alpha\}_{\alpha \in G}$ of $G$, the union $\bigcup_{\alpha \in G} U_\alpha \in G$ (3) G is closed under finite intersection: For any finite subcollection $\{U_1, U_2, ..., U_n\}$ of $G$, the intersection $\bigcap U_i \in G$ The pair $(X, G)$ is called the **topological space** $X$ with topology $G$. An **open set** in $X$ is a member of $G$. Let $(X, G_1)$ and $(Y, G_2)$ be topological spaces. A function $f : X \to Y$ is said to be **continuous** if $f^{-1}(V) \in G_1 \; \forall \; V \in G_2$. ### Sigma Algebras Let $X$ be a set with a collection $\mathcal{F}$ of subsets of $X$. $\mathcal{F}$ is a **$\sigma$-algebra** on $X$ iff: (1) $\varnothing, X \in \mathcal{F}$ (2) If $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$ (3) G is closed under countable union: For any countable subcollection $\{U_1, U_2, ..., U_n\}$ of $G$, the union $\bigcup U_i \in G$ Note: From (2) and (3), $\mathcal{F}$ is also closed under countable intersections. The pair $(X, \mathcal{F})$ is called a **measurable space** $X$ with $\sigma$-algebra $\mathcal{F}$. A **measurable set** in $X$ is a member of $\mathcal{F}$. Let $(X, \mathcal{F}_1)$ and $(Y, \mathcal{F}_2)$ be measurable spaces. A function $f : X \to Y$ is said to be **measurable** if $f^{-1}(V) \in \mathcal{F}_1 \; \forall \; V \in \mathcal{F}_2$. A measurable function is a **random variable** if its range is $\mathcal{R}$. ### Borel Sets For a topological space $\Omega$, the smallest $\sigma$-algebra that also contains all open sets in $\Omega$ is called a **Borel space** $\mathcal{B}$. The members of $\mathcal{B}$ are called **Borel sets**. $\mathcal{B(\Omega)} = (\Omega, \mathcal{B(\Omega, G)})$ A **measure** (positive) is a function $\mu$ defined on a $\sigma$-algebra $\mathcal{F}$ whose range is in $[0, \infty)$ and is countably additive. This means that if $A_i$ is a disjoint countable collection of members of $\mathcal{F}$, then: $\mu(\bigcup_i A_i) = \sum_i \mu(A_i) \qquad(\exists A_i: \mu(A_i) < \infty \iff \mu(\varnothing) = 0$) $\mathcal{(\Omega, F, \mu)}$ is known as a **measure space**. If $\mu(\Omega) = 1$ then $\mathcal{(\Omega, F, \mu)}$ is a **probability space** $\mathcal{(\Omega, F, P)}$ with event set $\Omega$ and probability measure $\mathcal{P}$ **Bayes Rule:** For any $A, B \in \mathcal{F}$, if $P(B) > 0$, $P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{P(A \cap B)}{P(B)}$ **Independence:** Two events are independent if $P(X \cup Y) = P(X)P(Y)$  ### Filtration E.g. Flip a coin $n$ times; $A_{k_1, k_2} = \mathcal{P}$(First k1, k2, ... tosses were H) $\mathcal{F_0} = \{\varnothing, \Omega\}$ $\mathcal{F_1} = \{\varnothing, \Omega, A_0, A_1\}$ $\mathcal{F_2} = \{\varnothing, \Omega, A_{00}, A_{01}, A_{10}, A_{11},$ $A_0, A_1, A_{00} \cup A_{10}, A_{11} \cup A_{01}, A_{00} \cup A_{11}, A_{10} \cup A_{01},$ $A_{00}^c, A_{01}^c, A_{10}^c, A_{11}^c\}$ ... $\mathcal{F_n} = 2^\Omega (\textrm{Power Set})$ Given a measurable space $(\Omega, \mathcal{F})$, a **filtration** is a sequence of sub-$\sigma$ algebras of $\mathcal{F}$ such that $\mathcal{F}_t \subseteq \mathcal{F}_{t+1} \; \forall \; t < n$. A sequence of random variables $(X_t)^n$ is **adapted** to filtration $F = (F_t)^n$ if $X_t$ is $F_t$-measurable at each $1 \le t \le n$. ### Lebesgue Integral Consider a measure space $(\Omega, \mathcal{F}, \mu)$.An indicator function $I_A(\omega)$ is 1 if $\omega \in A$ and zero otherwise. We define integral of $I_A(\omega)$ wrt the measure $\mu$ as follows: $\int_{\Omega} I_A(\omega)d\mu(\omega) = \mu(A)$ for $A \in \mathcal{F}$ A simple function is a measurable function $h$ on a measurable space $\Omega$ whose range consists of only finitely many points in $\mathcal{R}^+$: $h = \sum_{i=1}^n \alpha_i I_{A_i}$; $\alpha_i$ are distinct and $A_i = \{\omega h(\omega) = \alpha_i\}$ $\int h d\mu = \sum \alpha_i\mu(A_i)$ Let $X: \Omega \to \mathcal{R}^+$ be a measurable function on $(\Omega, \mathcal{F}, \mu)$ then we have the **Lebesgue integral**: $\int Xd\mu = sup\{\int hd\mu: h, \alpha \le h \le X\}$ If $\Omega = \mathcal{R}, \mathcal{F}$ is called a **Lebesgue $\sigma$-algebra** where the **Lebesgue measure** $\lambda$ is unique such that $\lambda((a,b]) = b-a$ Consider two measures $\lambda, \mu$ on $(\Omega, \mathcal{F})$. $\lambda$ is **absolutely continuous** with respect to $\mu$ if $\lambda(A) = 0 \; \forall \; \{A \in \mathcal{F} \mid \mu(A) = 0\}$ If $\lambda << \mu$ then there exists a function $h$ where $\int|h|d\mu \le \infty$ such that: $\lambda(A) = \int_A h d\mu = \int_Ad\lambda = I_Ad\lambda \; \forall \; A \in \mathcal{F}$ $h = \frac{d\lambda}{d\mu}$ is the **Radon-Nikodym derivative** and describes the change of measure density. E.g. Gaussian measure $(\mathcal{B,F})$ $\lambda(A) = \int_A\frac{1}{2\pi}e^{-\frac{x^2}{2}}d\mu$ (Here, $d\mu = dx$) The Gaussian measure is absolutely continuous wrt Lebesgue measure: $\frac{d\lambda}{d\mu} = \frac{1}{2\pi}e^{-\frac{x^2}{2}}$ (Density PDF) If $X: \Omega \to \mathcal{R}$ is a random variable on $(\mathcal{\Omega, F, P})$ then: $E[X] = \int Xd\mathcal{P}$ $(\mathcal{\Omega, F, P}) \to^X (\mathcal{R, B(R), P}_X)$ ### Conditional Expectation $Y = E[X|Z], Y(\omega) = E[X|Z](\omega)$, Y is defined on the $\sigma$-algebra that $Z$ is defined on. Consider $\mathcal{G}$ to be a sub-$\sigma$-algebra of $\mathcal{F}$. For a random variable $X$ on $(\mathcal{\Omega,F,P})$ with $E[|X|] < \infty$, there exists another random variable Y that is $\mathcal{G}-measurable$ and $E[|Y|] < \infty$, such that for $G \in \mathcal{G}$: $\int_G Y\mathcal{dP} = \int_G X\mathcal{dP}, Y = E[X|\mathcal{G}](G)$ Y is called a version of the conditional expectation and we have $Y = E[X|\mathcal{G}]$ a.s.