<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass{article}
\pagestyle{empty}
%\usepackage{xgreek,graphicx}
\usepackage{polyglossia}
\setdefaultlanguage{english}
\setotherlanguage{greek}
\setotherlanguage{russian}

\usepackage{fontspec}
\usepackage{unicode-math}


\RequirePackage{fontspec}
\RequirePackage{unicode-math}
\setmainfont[%
ItalicFont=NewCM10-BookItalic.otf,%
BoldFont=NewCM10-Bold.otf,%
BoldItalicFont=NewCM10-BoldItalic.otf,%
SmallCapsFeatures={Numbers=OldStyle}]{NewCM10-Book.otf}

\setsansfont[%
ItalicFont=NewCMSans10-BookOblique.otf,%
BoldFont=NewCMSans10-Bold.otf,%
BoldItalicFont=NewCMSans10-BoldOblique.otf,%
SmallCapsFeatures={Numbers=OldStyle}]{NewCMSans10-Book.otf}

\setmonofont[ItalicFont=NewCMMono10-BookItalic.otf,%
BoldFont=NewCMMono10-Bold.otf,%
BoldItalicFont=NewCMMono10-BoldOblique.otf,%
SmallCapsFeatures={Numbers=OldStyle}]{NewCMMono10-Book.otf}

\setmathfont{NewCMMath-Book.otf}

\newcommand{\tttextsc}[1]{{\ttscshape#1}}

\newtheorem{theorem}{Theorem}
\newtheorem{theoremg}[theorem]{Î˜ÎµÏŽÏÎ·Î¼Î±}
\newtheorem{theoremr}[theorem]{Ñ‚ÐµÐ¾Ñ€ÐµÐ¼Ð°}

\begin{document}

---A

\begin{theorem}[Dominated convergence of Lebesgue]
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}
\textsc{Proof}: The function $g-f_n$ is 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
$$\int f\geq \limsup \int f_n.$$

\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}
\textsc{Î‘Ï€ÏŒÎ´ÎµÎ¹Î¾Î·}: Î— ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· $g âˆ’ f_n$ ÎµÎ¯Î½Î±Î¹ 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±
$$\int f\geq \limsup \int f_n.$$

\selectlanguage{russian}
Thanks to Ð¡ÐµÑ€Ð³ÐµÐ¹ ÐœÐ°Ñ€Ñ‚Ñ‹Ð½Ð¾Ð² for the translation to Russian:
\begin{theoremr}
ÐŸÑ€ÐµÐ´Ð¿Ð¾Ð»Ð¾Ð¶Ð¸Ð¼, Ñ‡Ñ‚Ð¾ $g$ ÑÐ²Ð»ÑÐµÑ‚ÑÑ
Ð¸Ð½Ñ‚ÐµÐ³Ñ€Ð¸Ñ€ÑƒÐµÐ¼Ð¾Ð¹ Ñ„ÑƒÐ½ÐºÑ†Ð¸ÐµÐ¹, Ð¾Ð¿Ñ€ÐµÐ´ÐµÐ»ÐµÐ½Ð½Ð¾Ð¹ Ð½Ð° Ð¸Ð·Ð¼ÐµÑ€Ð¸Ð¼Ð¾Ð¼ Ð¼Ð½Ð¾Ð¶ÐµÑÑ‚Ð²Ðµ $E$, Ð¸
$(f_n)_{n\in\mathbb N}$ Ð¿Ñ€ÐµÐ´ÑÑ‚Ð°Ð²Ð»ÑÐµÑ‚ ÑÐ¾Ð±Ð¾Ð¹ Ð¿Ð¾ÑÐ»ÐµÐ´Ð¾Ð²Ð°Ñ‚ÐµÐ»ÑŒÐ½Ð¾ÑÑ‚ÑŒ Ð¸Ð·Ð¼ÐµÑ€Ð¸Ð¼Ð¾Ð¹ Ñ„ÑƒÐ½ÐºÑ†Ð¸Ð¸, Ñ‚Ð°Ðº Ñ‡Ñ‚Ð¾
   $|f_n|\leq g$. Ð•ÑÐ»Ð¸ $f$ ÑÐ²Ð»ÑÐµÑ‚ÑÑ Ñ„ÑƒÐ½ÐºÑ†Ð¸ÐµÐ¹, Ñ‚Ð°Ðº Ñ‡Ñ‚Ð¾ $f_n\to f$ Ð¿Ð¾Ñ‡Ñ‚Ð¸ Ð²ÐµÐ·Ð´Ðµ,
   Ñ‚Ð¾Ð³Ð´Ð°
$$\lim \int f_n =\int f.$$
\end{theoremr}

\newpage


\selectlanguage{english}
\sffamily

\begin{theorem}[Dominated convergence of Lebesgue]
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}
\textsc{Proof}: The function $g-f_n$ is 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
$$\int f\geq \limsup \int f_n.$$

\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}
\textsc{Î‘Ï€ÏŒÎ´ÎµÎ¹Î¾Î·}: Î— ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· $g âˆ’ f_n$ ÎµÎ¯Î½Î±Î¹ 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±
$$\int f\geq \limsup \int f_n.$$

\selectlanguage{russian}

Thanks to Ð¡ÐµÑ€Ð³ÐµÐ¹ ÐœÐ°Ñ€Ñ‚Ñ‹Ð½Ð¾Ð² for the translation to Russian:
\begin{theoremr}
ÐŸÑ€ÐµÐ´Ð¿Ð¾Ð»Ð¾Ð¶Ð¸Ð¼, Ñ‡Ñ‚Ð¾ $g$ ÑÐ²Ð»ÑÐµÑ‚ÑÑ
Ð¸Ð½Ñ‚ÐµÐ³Ñ€Ð¸Ñ€ÑƒÐµÐ¼Ð¾Ð¹ Ñ„ÑƒÐ½ÐºÑ†Ð¸ÐµÐ¹, Ð¾Ð¿Ñ€ÐµÐ´ÐµÐ»ÐµÐ½Ð½Ð¾Ð¹ Ð½Ð° Ð¸Ð·Ð¼ÐµÑ€Ð¸Ð¼Ð¾Ð¼ Ð¼Ð½Ð¾Ð¶ÐµÑÑ‚Ð²Ðµ $E$, Ð¸
$(f_n)_{n\in\mathbb N}$ Ð¿Ñ€ÐµÐ´ÑÑ‚Ð°Ð²Ð»ÑÐµÑ‚ ÑÐ¾Ð±Ð¾Ð¹ Ð¿Ð¾ÑÐ»ÐµÐ´Ð¾Ð²Ð°Ñ‚ÐµÐ»ÑŒÐ½Ð¾ÑÑ‚ÑŒ Ð¸Ð·Ð¼ÐµÑ€Ð¸Ð¼Ð¾Ð¹ Ñ„ÑƒÐ½ÐºÑ†Ð¸Ð¸, Ñ‚Ð°Ðº Ñ‡Ñ‚Ð¾
   $|f_n|\leq g$. Ð•ÑÐ»Ð¸ $f$ ÑÐ²Ð»ÑÐµÑ‚ÑÑ Ñ„ÑƒÐ½ÐºÑ†Ð¸ÐµÐ¹, Ñ‚Ð°Ðº Ñ‡Ñ‚Ð¾ $f_n\to f$ Ð¿Ð¾Ñ‡Ñ‚Ð¸ Ð²ÐµÐ·Ð´Ðµ,
   Ñ‚Ð¾Ð³Ð´Ð°
$$\lim \int f_n =\int f.$$
\end{theoremr}




\newpage


\selectlanguage{english}
\ttfamily

\begin{theorem}[Dominated convergence of Lebesgue]
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}
\textsc{Proof}: The function $g-f_n$ is 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
$$\int f\geq \limsup \int f_n.$$


\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}
\textsc{Î‘Ï€ÏŒÎ´ÎµÎ¹Î¾Î·}: Î— ÏƒÏ…Î½Î¬ÏÏ„Î·ÏƒÎ· $g âˆ’ f_n$ ÎµÎ¯Î½Î±Î¹ 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±
$$\int f\geq \limsup \int f_n.$$


\selectlanguage{russian}
Thanks to Ð¡ÐµÑ€Ð³ÐµÐ¹ ÐœÐ°Ñ€Ñ‚Ñ‹Ð½Ð¾Ð² for the translation to Russian:
\begin{theoremr}
ÐŸÑ€ÐµÐ´Ð¿Ð¾Ð»Ð¾Ð¶Ð¸Ð¼, Ñ‡Ñ‚Ð¾ $g$ ÑÐ²Ð»ÑÐµÑ‚ÑÑ
Ð¸Ð½Ñ‚ÐµÐ³Ñ€Ð¸Ñ€ÑƒÐµÐ¼Ð¾Ð¹ Ñ„ÑƒÐ½ÐºÑ†Ð¸ÐµÐ¹, Ð¾Ð¿Ñ€ÐµÐ´ÐµÐ»ÐµÐ½Ð½Ð¾Ð¹ Ð½Ð° Ð¸Ð·Ð¼ÐµÑ€Ð¸Ð¼Ð¾Ð¼ Ð¼Ð½Ð¾Ð¶ÐµÑÑ‚Ð²Ðµ $E$, Ð¸
$(f_n)_{n\in\mathbb N}$ Ð¿Ñ€ÐµÐ´ÑÑ‚Ð°Ð²Ð»ÑÐµÑ‚ ÑÐ¾Ð±Ð¾Ð¹ Ð¿Ð¾ÑÐ»ÐµÐ´Ð¾Ð²Ð°Ñ‚ÐµÐ»ÑŒÐ½Ð¾ÑÑ‚ÑŒ Ð¸Ð·Ð¼ÐµÑ€Ð¸Ð¼Ð¾Ð¹ Ñ„ÑƒÐ½ÐºÑ†Ð¸Ð¸, Ñ‚Ð°Ðº Ñ‡Ñ‚Ð¾
   $|f_n|\leq g$. Ð•ÑÐ»Ð¸ $f$ ÑÐ²Ð»ÑÐµÑ‚ÑÑ Ñ„ÑƒÐ½ÐºÑ†Ð¸ÐµÐ¹, Ñ‚Ð°Ðº Ñ‡Ñ‚Ð¾ $f_n\to f$ Ð¿Ð¾Ñ‡Ñ‚Ð¸ Ð²ÐµÐ·Ð´Ðµ,
   Ñ‚Ð¾Ð³Ð´Ð°
$$\lim \int f_n =\int f.$$
\end{theoremr}


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