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author | Karl Berry <karl@freefriends.org> | 2019-12-20 22:21:32 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2019-12-20 22:21:32 +0000 |
commit | f27cb1d537a34677b1291d5365c80cde1416417c (patch) | |
tree | 0e2d0e89e70b388b3f45fe8e81491a09cdca51e8 /Master/texmf-dist/doc/latex/fontsetup/fspsample.tex | |
parent | 42fff9c15d299e41425b1aef482186e3cd2f4d9a (diff) |
fontsetup (20dec19)
git-svn-id: svn://tug.org/texlive/trunk@53195 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex/fontsetup/fspsample.tex')
-rw-r--r-- | Master/texmf-dist/doc/latex/fontsetup/fspsample.tex | 72 |
1 files changed, 72 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/latex/fontsetup/fspsample.tex b/Master/texmf-dist/doc/latex/fontsetup/fspsample.tex new file mode 100644 index 00000000000..b9783698540 --- /dev/null +++ b/Master/texmf-dist/doc/latex/fontsetup/fspsample.tex @@ -0,0 +1,72 @@ +\documentclass{article} +\pagestyle{empty} +\usepackage{xgreek,graphicx} +\pagestyle{empty} +%\usepackage[greek]{babel} +%\usepackage[utf8x]{inputenc} +%\usepackage{amsfonts} + +\usepackage[default]{fontsetup} +%\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[neoeuler]{fontsetup} +%\usepackage[libertinus]{fontsetup} + + + + +\newtheorem{theorem}{Theorem} +\newtheorem{theoremg}[theorem]{Θεώρημα} + + +\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 hat + $(f_n)_{n\in\mathbb N}$ is a sequence of mea\-sur\-able function 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} +\textit{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 +$$\int f\geq \limsup \int f_n.$$ +\par +\begin{theoremg}[Κυριαρχημένης σύγκλισης του Lebesgue] + Έστω ότι +η $g$ είναι μια ολοκληρώσιμη συνάρτηση ορισμένη στο μετρήσιμο σύνολο +$E$ και η $(f_n)_{n\in\mathbb N}$ είναι μια ακολουθία μετρήσιμων συναρτήσεων ώστε +$|f_n| ≤ g$. Υποθέτουμε ότι υπάρχει μια συνάρτηση $f$ +ώστε η $(f_n)_{n\in\mathbb N}$ να +τείνει στην $f$ σχεδόν παντού. Τότε +$$\lim \int f_n =\int f.$$ +\end{theoremg} +\textit{Απόδειξη}: Η συνάρτηση $g − f_n$ είναι μη αρνητική και άρα από +το Λήμμα του Fatou ισχύει +$\int (f-g) ≤ \liminf \int (g-f_n)$. Επειδή +$|f| ≤ g$ και $|f_n| ≤g$ οι $f$ και $f_n$ είναι ολοκληρώσιμες, έχουμε +$$\int g −\int f ≤ \int g − \limsup\int f_n,$$ +άρα +$$\int f\geq \limsup \int f_n.$$ + + + + + +\end{document} |