From 2519a62567438ed1d8bb680f329bcc13cc21dfd8 Mon Sep 17 00:00:00 2001 From: Zentrik Date: Mon, 27 May 2024 21:19:02 +0100 Subject: [PATCH] P&M: More typos --- ProbAndMeasure/02_measurable_functions.tex | 11 +++++++---- ProbAndMeasure/probmeasure.pdf | Bin 397815 -> 397974 bytes 2 files changed, 7 insertions(+), 4 deletions(-) diff --git a/ProbAndMeasure/02_measurable_functions.tex b/ProbAndMeasure/02_measurable_functions.tex index 014ec95..629c18c 100644 --- a/ProbAndMeasure/02_measurable_functions.tex +++ b/ProbAndMeasure/02_measurable_functions.tex @@ -365,7 +365,7 @@ \subsection{Convergence of measurable functions} \begin{align*} \mu\qty(\abs{f_n} > \frac{1}{k}) \to 0. \end{align*} - So we can choose $n_k$ s.t. $\mu\qty(\abs{f_n} > \frac{1}{k}) \leq \frac{1}{k^2}$. + So we can choose $n_k$ s.t. $\mu\qty(\abs{f_{n_k}} > \frac{1}{k}) \leq \frac{1}{k^2}$. We can choose $n_{k+1}$ in the same way s.t. $n_{k+1} > n_k$. So we get a subsequence $n_k$ s.t. $\mu\qty(\abs{f_{n_k}} > \frac{1}{k}) < \frac{1}{k^2}$. Also $\sum_k \frac{1}{k^2} < \infty$, so $\sum_k \mu\qty(\abs{f_{n_k}} > \frac{1}{k}) < \infty$. @@ -408,9 +408,12 @@ \subsection{Convergence of measurable functions} \end{example} \subsection{Kolmogorov's zero-one law} -Let $(X_n)_{n \in \mathbb N}$ be a sequence of r.v.s. -We can define $\mathcal T_n = \sigma(X_{n+1}, X_{n+2}, \dots)\footnote{The smallest $\sigma$-algebra s.t. $X_{n+1}, \dots$ are measurable.}$. -Let $\mathcal T = \bigcap_{n \in \mathbb N} \mathcal T_n$ be the \vocab{tail $\sigma$-algebra}, which contains all events in $\mathcal F$ that depend only on the `limiting behaviour' of $(X_n)$. + +\begin{definition}[Tail $\sigma$-Algebra] + Let $(X_n)_{n \in \mathbb N}$ be a sequence of r.v.s. + We can define $\mathcal T_n = \sigma(X_{n+1}, X_{n+2}, \dots)\footnote{The smallest $\sigma$-algebra s.t. $X_{n+1}, \dots$ are measurable.}$. + Let $\mathcal T = \bigcap_{n \in \mathbb N} \mathcal T_n$ be the \vocab{tail $\sigma$-algebra}, which contains all events in $\mathcal F$ that depend only on the `limiting behaviour' of $(X_n)$. +\end{definition} \begin{theorem}[Kolmogorov 0-1 Law] Let $(X_n)_{n \in \mathbb N}$ be a sequence of independent r.v.s. diff --git a/ProbAndMeasure/probmeasure.pdf b/ProbAndMeasure/probmeasure.pdf index 9c2790bb65f5c0db1235c695f93feb21cd027b1b..65f0e6f87f3ebcf5022f9b6a31d78aa0fbee2c06 100644 GIT binary patch delta 111190 zcmV(~K+nJTqZpQ@7=W|^Jam6GGCn>Eb98cLVQmU{+SOazjvTiUea~04^CTe{n)ge* z9>fl80|{W*jvfR=d02T@Mq*2_9S70p=M-;Eve|Rlm7D~^YG*lQlPp%9syeJ*epKaa*)z9r{waw zwg|}tCj4Rf%kuRD=2%Ar8@m7Fh_NxP3TyCoU5plni4YbkEHhdyKi@B}9=DjC-=98P zKKxvSMPM;Jh~?RFH;9>MBn1))$t%TiALYk7Vsm}cH$OQ zYU6k4WjOVsjfPc_apUnh`0+CV2(FnEHZSY9vvp%x!sQMh=MH~DG%|u#2W}S4WH7Op zc$$yyr-iBD4QxVM1D$j@E6tQpunFm)Nxa*1FmqeuEW%F`Wzt|GZQ)aXFYR_P&3h93xGZ`G}k+*t5XzocbTw2ba zkw{KJnSGUU*p}kMwoG%7npr6q z$s}SL>Lh2HO*#=ZTNe0-HXpPiyHoyXb+ttCSWBcy!b424lPvA3=+y7TY)b$#4Zqr2 zYYhq#!kFVhnBv!L*@`KfyYE{4QARkD$?uulxVe9WLLlg2ya(8Mkl}Jzz!@v?e#D>$ 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