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authorNorbert Preining <norbert@preining.info>2020-09-26 03:03:17 +0000
committerNorbert Preining <norbert@preining.info>2020-09-26 03:03:17 +0000
commit66ebd4914b367eb00a52ce5d5bb51f695101931b (patch)
treee5e4a5148bbf213f7ecc242d333bbde165d41e3b /macros/latex/contrib/fontsetup/doc/fspsample.tex
parent56c9134f4aa02292ed17e726efd8e493b90e6011 (diff)
CTAN sync 202009260303
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-%% after xelatex run
-%% pdfcrop fspsample.pdf output.pdf
-%% to crop it
-\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}
-%\usepackage[olddefault]{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 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}
-\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}