<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[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]{ÎÎµĎŽĎÎ·ÎĽÎ±}

\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}[ÎšĎ…ĎÎąÎ±ĎĎ‡Î·ÎĽÎ­Î˝Î·Ď‚ ĎĎŤÎłÎşÎ»ÎąĎÎ·Ď‚ Ď„ÎżĎ… Lebesgue]
  ÎĎĎ„Ď‰ ĎŚĎ„Îą
Î· $g$ ÎµÎŻÎ˝Î±Îą ÎĽÎąÎ± ÎżÎ»ÎżÎşÎ»Î·ĎĎŽĎÎąÎĽÎ· ĎĎ…Î˝Î¬ĎĎ„Î·ĎÎ· ÎżĎÎąĎÎĽÎ­Î˝Î· ĎĎ„Îż ÎĽÎµĎ„ĎÎ®ĎÎąÎĽÎż ĎĎŤÎ˝ÎżÎ»Îż
$E$ ÎşÎ±Îą Î· $(\,f_n)_{n\in\mathbb N}$ ÎµÎŻÎ˝Î±Îą ÎĽÎąÎ± Î±ÎşÎżÎ»ÎżĎ…Î¸ÎŻÎ± ÎĽÎµĎ„ĎÎ®ĎÎąÎĽĎ‰Î˝ ĎĎ…Î˝Î±ĎĎ„Î®ĎÎµĎ‰Î˝ ĎŽĎĎ„Îµ
$|\,f_n| â‰¤ g$. ÎĄĎ€ÎżÎ¸Î­Ď„ÎżĎ…ÎĽÎµ ĎŚĎ„Îą Ď…Ď€Î¬ĎĎ‡ÎµÎą ÎĽÎąÎ± ĎĎ…Î˝Î¬ĎĎ„Î·ĎÎ· $f$
ĎŽĎĎ„Îµ Î·  $(f_n)_{n\in\mathbb N}$ Î˝Î±
Ď„ÎµÎŻÎ˝ÎµÎą ĎĎ„Î·Î˝ $f$ ĎĎ‡ÎµÎ´ĎŚÎ˝ Ď€Î±Î˝Ď„ÎżĎŤ. Î¤ĎŚĎ„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¬Î˝Ď„Î± ĎÎµ ÎłĎÎ±ÎĽÎĽÎ±Ď„ÎżĎÎµÎąĎÎ¬ Sans}
\end{wrapfigure}

\textit{Î‘Ď€ĎŚÎ´ÎµÎąÎľÎ·}: Î— ĎĎ…Î˝Î¬ĎĎ„Î·ĎÎ· $g â’ f_n$ ÎµÎŻÎ˝Î±Îą \textit{ÎĽÎ· Î±ĎÎ˝Î·Ď„ÎąÎşÎ®} ÎşÎ±Îą Î¬ĎÎ± Î±Ď€ĎŚ
Ď„Îż Î›Î®ÎĽÎĽÎ± Ď„ÎżĎ… Fatou ÎąĎĎ‡ĎŤÎµÎą
$\int (f-g) â‰¤ \liminf \int (g-f_n)$. Î•Ď€ÎµÎąÎ´Î®
$|\,f\,| â‰¤ g$ ÎşÎ±Îą $|\,f_n| â‰¤g$ ÎżÎą $f$ ÎşÎ±Îą $f_n$ ÎµÎŻÎ˝Î±Îą ÎżÎ»ÎżÎşÎ»Î·ĎĎŽĎÎąÎĽÎµĎ‚, Î­Ď‡ÎżĎ…ÎĽÎµ
$$\int g â’\int f\, â‰¤ \int g â’ \limsup\int f_n,$$
Î¬ĎÎ±
$$\int f\,\geq \limsup \int f_n.$$





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