<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%% after xelatex run
%% pdfcrop fspsample.pdf output.pdf
%% to crop it
\documentclass{article}
\pagestyle{empty}
\usepackage{graphicx,wrapfig,xcolor}
\pagestyle{empty}
%\usepackage[greek]{babel}
%\usepackage[utf8x]{inputenc}
%\usepackage{amsfonts}

\usepackage{polyglossia}
\setdefaultlanguage{english}
\setotherlanguage{greek}

\usepackage[default]{fontsetup}%upint, newcmbb
%\usepackage[sansdefault]{fontsetup}%upint, newcmbb
%\usepackage[gfsartemisia]{fontsetup}
%\usepackage[gfsdidot]{fontsetup}
%\usepackage[gfsdidotclassic]{fontsetup}
%\usepackage[gfsneohellenic]{fontsetup}
%\usepackage[cambria]{fontsetup}
%\usepackage[lucida]{fontsetup}
%\usepackage[kerkis]{fontsetup}
%\usepackage[fira]{fontsetup}
%\usepackage[times]{fontsetup}
%\usepackage[palatino]{fontsetup}
%\usepackage[stixtwo]{fontsetup}
%\usepackage[neokadmus]{fontsetup}
%\usepackage[msgaramond]{fontsetup}
%\usepackage[ebgaramond]{fontsetup}
%\usepackage[minion]{fontsetup}
%\usepackage[euler]{fontsetup}
%\usepackage[neoeuler]{fontsetup}
%\usepackage[libertinus]{fontsetup}
%\usepackage[olddefault]{fontsetup}%upint, newcmbb
%\usepackage[concrete]{fontsetup}
%\usepackage[talos]{fontsetup}
%\usepackage[oldstandard]{fontsetup}
%\usepackage[xcharter,upint]{fontsetup}
%\usepackage[erewhon]{fontsetup}

\newtheorem{theorem}{Theorem}
\newtheorem{theoremg}[theorem]{Î˜ÎµÏŽÏÎ·Î¼Î±}

\makeatletter
\long\def\@makecaption#1#2{%
  \vskip\abovecaptionskip
  \sbox\@tempboxa{\sffamily #1: #2}%
  \ifdim \wd\@tempboxa &gt;\hsize
   {\sffamily #1: #2}\par
  \else
    \global \@minipagefalse
    \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
  \fi
  \vskip\belowcaptionskip}
\makeatother




\begin{document}

\begin{theorem}[Dominated convergence of Lebesgue]
%Let $g$ be an
Assume that $g$ is an
in\-te\-grable func\-tion defined on the measurable set $E$ and that
  $(\,f_n)_{n\in\mathbb N}$ is a sequence of mea\-sur\-able functions so that
  $|\,f_n|\leq g$. If $f$ is a function so that $f_n\to f$ almost everywhere
  then $$\lim_{n\to\infty}\int f_n=\int f.$$
\end{theorem}

\begin{wrapfigure}[4]{o}{0.22\textwidth}\vspace*{-1.8\baselineskip}
\begin{center}
  {\Huge\textsf{\color{lightgray}ABC}}
\end{center}
\caption{Caption in Sans fonts}
\end{wrapfigure}
\textit{Proof}: The function $g-f_n$ is \textit{non-negative} and thus from Fatou lemma
we have that $\int(g-f\,)\leq\liminf\int(g-f_n)$. Since $|\,f\,|\leq g$ and
$|\,f_n|\leq g$ the  functions $f$ and $f_n$ are integrable and we have
$$\int g-\int f\,\leq \int g-\limsup\int f_n,$$ so
$$\int f\,\geq \limsup \int f_n.$$
\par

\selectlanguage{greek}


\begin{theoremg}[ÎšÏ…ÏÎ¹Î±ÏÏ‡Î·Î¼Î­Î½Î·Ï‚ ÏƒÏÎ³ÎºÎ»Î¹ÏƒÎ·Ï‚ Ï„Î¿Ï… Lebesgue]
  ÎˆÏƒÏ„Ï‰ ÏŒÏ„Î¹
Î· $g$ ÎµÎ¯Î½Î±Î¹ Î¼Î¹Î± Î¿Î»Î¿ÎºÎ»Î·ÏÏŽÏƒÎ¹Î¼Î· ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· Î¿ÏÎ¹ÏƒÎ¼Î­Î½Î· ÏƒÏ„Î¿ Î¼ÎµÏ„ÏÎ®ÏƒÎ¹Î¼Î¿ ÏƒÏÎ½Î¿Î»Î¿
$E$ ÎºÎ±Î¹ Î· $(\,f_n)_{n\in\mathbb N}$ ÎµÎ¯Î½Î±Î¹ Î¼Î¹Î± Î±ÎºÎ¿Î»Î¿Ï…Î¸Î¯Î± Î¼ÎµÏ„ÏÎ®ÏƒÎ¹Î¼Ï‰Î½ ÏƒÏ…Î½Î±ÏÏ„Î®ÏƒÎµÏ‰Î½ ÏŽÏƒÏ„Îµ
$|\,f_n| â‰¤ g$. Î¥Ï€Î¿Î¸Î­Ï„Î¿Ï…Î¼Îµ ÏŒÏ„Î¹ Ï…Ï€Î¬ÏÏ‡ÎµÎ¹ Î¼Î¹Î± ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· $f$
ÏŽÏƒÏ„Îµ Î·  $(f_n)_{n\in\mathbb N}$ Î½Î±
Ï„ÎµÎ¯Î½ÎµÎ¹ ÏƒÏ„Î·Î½ $f$ ÏƒÏ‡ÎµÎ´ÏŒÎ½ Ï€Î±Î½Ï„Î¿Ï. Î¤ÏŒÏ„Îµ
$$\lim \int f_n =\int f.$$
\end{theoremg}


\begin{wrapfigure}[4]{o}{0.22\textwidth}\vspace*{-1.8\baselineskip}
\begin{center}
  {\Huge\textsf{\color{lightgray}Î“Î¨Î©}}
\end{center}
\caption{Î›ÎµÎ¶Î¬Î½Ï„Î± ÏƒÎµ Î³ÏÎ±Î¼Î¼Î±Ï„Î¿ÏƒÎµÎ¹ÏÎ¬ Sans}
\end{wrapfigure}

\textit{Î‘Ï€ÏŒÎ´ÎµÎ¹Î¾Î·}: Î— ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· $g âˆ’ f_n$ ÎµÎ¯Î½Î±Î¹ \textit{Î¼Î· Î±ÏÎ½Î·Ï„Î¹ÎºÎ®} ÎºÎ±Î¹ Î¬ÏÎ± Î±Ï€ÏŒ
Ï„Î¿ Î›Î®Î¼Î¼Î± Ï„Î¿Ï… Fatou Î¹ÏƒÏ‡ÏÎµÎ¹
$\int (f-g) â‰¤ \liminf \int (g-f_n)$. Î•Ï€ÎµÎ¹Î´Î®
$|\,f\,| â‰¤ g$ ÎºÎ±Î¹ $|\,f_n| â‰¤g$ Î¿Î¹ $f$ ÎºÎ±Î¹ $f_n$ ÎµÎ¯Î½Î±Î¹ Î¿Î»Î¿ÎºÎ»Î·ÏÏŽÏƒÎ¹Î¼ÎµÏ‚, Î­Ï‡Î¿Ï…Î¼Îµ
$$\int g âˆ’\int f\, â‰¤ \int g âˆ’ \limsup\int f_n,$$
Î¬ÏÎ±
$$\int f\,\geq \limsup \int f_n.$$





\end{document}
</pre></body></html>