<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% !TeX root = ../main.tex

\chapter{ć°ĺ­Ś}

\section{ć°ĺ­ŚçŹŚĺˇĺĺŹĺź}
ăĺä˝č§čăć˛ĄćĺŻšć°ĺ­ŚçŹŚĺˇĺĺŹĺźĺĺˇä˝čŚćąďźčżéćäťŹčŽŠć°ĺ­ŚçŹŚĺˇéľĺžŞ GB/T 3102.11â1993ăçŠçç§ĺ­ŚĺććŻä¸­ä˝żç¨çć°ĺ­ŚçŹŚĺˇă
\footnote{ĺ GB 3102.11â1993ďźčŞ 2017 ĺš´ 3 ć 23 ćĽčľˇďźčŻĽć&nbsp;ĺč˝Źä¸şć¨čć§ć&nbsp;ĺă}ă
čŻĽć&nbsp;ĺĺç§éçşł ISO 31-11:1992 \footnote{çŽĺĺˇ˛ć´ć°ä¸ş ISO 80000-2:2019ă}ďź
ä˝ćŻä¸ \TeX{} éťčŽ¤ççžĺ˝ć°ĺ­Śĺ­ŚäźďźAMSďźççŹŚĺˇäš&nbsp;ćŻććĺşĺŤă
ĺˇä˝ĺ°ćĽčŻ´ä¸ťčŚćäťĽä¸ĺˇŽĺźďź
\begin{enumerate}
  \item ĺ¤§ĺĺ¸čĺ­ćŻéťčŽ¤ä¸şćä˝ďźĺŚ
    \begin{equation*}
      \Gamma \Delta \Theta \Lambda \Xi \Pi \Sigma \Upsilon \Phi \Psi \Omega.
    \end{equation*}
    ćł¨ććéĺ˘éçŹŚĺˇ $\increment$ ĺşĺŽä˝żç¨ć­Łä˝ďźć¨Ąćżćäžäş \cs{increment} ĺ˝äť¤ă
  \item ĺ°äşç­äşĺˇĺĺ¤§äşç­äşĺˇä˝żç¨ĺžćçĺ­ĺ˝˘ $\le$ă$\ge$ă
  \item ç§Żĺĺˇä˝żç¨ć­Łä˝ďźćŻĺŚ $\int$ă$\oint$ă
  \item čĄé´ĺŹĺźç§Żĺĺˇçä¸ä¸éä˝äşç§Żĺĺˇçä¸ä¸ä¸¤çŤŻďźćŻĺŚ
    \begin{equation*}
      \int_a^b f(x) \dif x.
    \end{equation*}
    čĄĺĺŹĺźä¸şäşçé˘ççžč§ďźçťä¸ĺąĺłäž§ďźĺŚ $\int_a^b f(x) \dif x$ ă
  \item
    ĺĺžŽĺçŹŚĺˇ $\partial$ ä˝żç¨ć­Łä˝ă
  \item
    ççĽĺˇ \cs{dots} ćç§ä¸­ćçäš&nbsp;ćŻĺşĺŽĺąä¸­ďźćŻĺŚ
    \begin{equation*}
      1, 2, \dots, n \quad 1 + 2 + \dots + n.
    \end{equation*}
  \item
    ĺŽé¨ $\Re$ ĺčé¨ $\Im$ çĺ­ä˝ä˝żç¨ç˝éŠŹä˝ă
\end{enumerate}

äťĽä¸ć°ĺ­ŚçŹŚĺˇć&nbsp;ˇĺźçĺˇŽĺźĺŻäťĽĺ¨ć¨Ąćżä¸­çťä¸čŽžç˝Žă
ä˝ćŻčżćä¸äşéčŚç¨ćˇĺ¨ĺä˝ćśčżčĄĺ¤çďź
\begin{enumerate}
  \item ć°ĺ­Śĺ¸¸ć°ĺçšćŽĺ˝ć°ĺç¨ć­Łä˝ďźĺŚ
    \begin{equation*}
      \uppi = 3.14\dots; \quad
      \symup{i}^2 = -1; \quad
      \symup{e} = \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n.
    \end{equation*}
  \item ĺžŽĺĺˇä˝żç¨ć­Łä˝ďźćŻĺŚ $\dif y / \dif x$ă
  \item ĺéăçŠéľĺĺź&nbsp;éç¨ç˛ćä˝ďź\cs{symbf}ďźďźĺŚ $\symbf{x}$ă$\symbf{\Sigma}$ă$\symbfsf{T}$ă
  \item čŞçśĺŻšć°ç¨ $\ln x$ ä¸ç¨ $\log x$ă
\end{enumerate}

ć¨Ąćżä¸­ä˝żç¨ \pkg{unicode-math} ĺŽĺéç˝Žć°ĺ­Śĺ­ä˝ă
čŻĽĺŽĺä¸äź&nbsp;çťç \pkg{amsfonts}ă\pkg{amssymb}ă\pkg{bm}ă
\pkg{mathrsfs}ă\pkg{upgreek} ç­ĺŽĺ\emph{ä¸}ĺźĺŽšă
ćŹć¨Ąćżä˝äşĺ¤çďźç¨ćˇĺŻäťĽç´ćĽä˝żç¨ \cs{bm}, \cs{mathscr},
\cs{upGamma} ç­ĺ˝äť¤ă
ĺłäşć°ĺ­ŚçŹŚĺˇć´ĺ¤çç¨ćłďźĺč§ \pkg{unicode-math} ĺŽĺçä˝żç¨čŻ´ćĺçŹŚĺˇĺčĄ¨
\pkg{unimath-symbols}ă



\section{éĺĺä˝}

ĺŽĺ \pkg{siunitx} ćäžäşć´ĺĽ˝çć°ĺ­ĺĺä˝ćŻćďź
\begin{itemize}
  \item \num{12345.67890}
  \item \num{.3e45}
  \item \si{kg.m.s^{-1}}
  \item \si{\micro\meter} $\si{\micro\meter}$
  \item \si{\ohm} $\si{\ohm}$
  \item \numlist{10;20}
  \item \numlist{10;20;30}
  \item \SIlist{0.13;0.67;0.80}{\milli\metre}
  \item \numrange{10}{20}
  \item \SIrange{10}{20}{\degreeCelsius}
\end{itemize}



\section{ĺŽçĺčŻć}

ç¤şäžćäťśä¸­ä˝żç¨ \pkg{amsthm} ĺŽĺéç˝ŽäşĺŽçăĺźçĺčŻćç­çŻĺ˘ă
ç¨ćˇäšĺŻäťĽä˝żç¨ \pkg{ntheorem} ĺŽĺă\par

ĺŹĺźĺźç¨ç¤şäžďźč§ĺź~\ref{eq:example}
\begin{definition}
  If the integral of function $f$ is measurable and non-negative, we define
  its (extended) \textbf{Lebesgue integral} by
  \begin{equation}
    \int f = \sup_g \int g,
  \end{equation}
  where the supremum is taken over all measurable functions $g$ such that
  $0 \le g \le f$, and where $g$ is bounded and supported on a set of
  finite measure.
\end{definition}

\begin{assumption}
The communication graph is strongly connected.
\end{assumption}

\begin{example}
  Simple examples of functions on $\mathbb{R}^d$ that are integrable
  (or non-integrable) are given by
  \begin{equation}
    f_a(x) =
    \begin{cases}
      |x|^{-a} &amp; \text{if } |x| \le 1, \\
      0        &amp; \text{if } x &gt; 1.
    \end{cases}
  \end{equation}
  \begin{equation}
    \label{eq:example}
    F_a(x) = \frac{1}{1 + |x|^a}, \qquad \text{all } x \in \mathbb{R}^d.
  \end{equation}
  Then $f_a$ is integrable exactly when $a &lt; d$, while $F_a$ is integrable
  exactly when $a &gt; d$.
\end{example}

\begin{lemma}[Fatou]
  Suppose $\{f_n\}$ is a sequence of measurable functions with $f_n \geq 0$.
  If $\lim_{n \to \infty} f_n(x) = f(x)$ for a.e. $x$, then
  \begin{equation}
    \int f \le \liminf_{n \to \infty} \int f_n.
  \end{equation}
\end{lemma}

\begin{remark}
  We do not exclude the cases $\int f = \infty$,
  or $\liminf_{n \to \infty} f_n = \infty$.
\end{remark}

\begin{corollary}
  Suppose $f$ is a non-negative measurable function, and $\{f_n\}$ a sequence
  of non-negative measurable functions with
  $f_n(x) \le f(x)$ and $f_n(x) \to f(x)$ for almost every $x$. Then
  \begin{equation}
    \lim_{n \to \infty} \int f_n = \int f.
  \end{equation}
\end{corollary}

\begin{proposition}
  Suppose $f$ is integrable on $\mathbb{R}^d$. Then for every $\epsilon &gt; 0$:
  \begin{enumerate}
    \renewcommand{\theenumi}{\roman{enumi}}
    \item There exists a set of finite measure $B$ (a ball, for example) such
      that
      \begin{equation}
        \int_{B^c} |f| &lt; \epsilon.
      \end{equation}
    \item There is a $\delta &gt; 0$ such that
      \begin{equation}
        \int_E |f| &lt; \epsilon \qquad \text{whenever } m(E) &lt; \delta.
      \end{equation}
  \end{enumerate}
\end{proposition}

\begin{theorem}
  Suppose $\{f_n\}$ is a sequence of measurable functions such that
  $f_n(x) \to f(x)$ a.e. $x$, as $n$ tends to infinity.
  If $|f_n(x)| \le g(x)$, where $g$ is integrable, then
  \begin{equation}
    \int |f_n - f| \to 0 \qquad \text{as } n \to \infty,
  \end{equation}
  and consequently
  \begin{equation}
    \int f_n \to \int f \qquad \text{as } n \to \infty.
  \end{equation}
\end{theorem}

\begin{proof}
  Trivial.
\end{proof}

\newtheorem*{axiomofchoice}{Axiom of choice}
\begin{axiomofchoice}
  Suppose $E$ is a set and ${E_\alpha}$ is a collection of
  non-empty subsets of $E$. Then there is a function $\alpha
  \mapsto x_\alpha$ (a ``choice function'') such that
  \begin{equation}
    x_\alpha \in E_\alpha,\qquad \text{for all }\alpha.
  \end{equation}
\end{axiomofchoice}

\newtheorem{observation}{Observation}
\begin{observation}
  Suppose a partially ordered set $P$ has the property
  that every chain has an upper bound in $P$. Then the
  set $P$ contains at least one maximal element.
\end{observation}
\begin{proof}[A concise proof]
  Obvious.
\end{proof}
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