diff options
author | Norbert Preining <norbert@preining.info> | 2020-09-26 03:03:17 +0000 |
---|---|---|
committer | Norbert Preining <norbert@preining.info> | 2020-09-26 03:03:17 +0000 |
commit | 66ebd4914b367eb00a52ce5d5bb51f695101931b (patch) | |
tree | e5e4a5148bbf213f7ecc242d333bbde165d41e3b /macros/latex/contrib/fontsetup/doc/fspsample.tex | |
parent | 56c9134f4aa02292ed17e726efd8e493b90e6011 (diff) |
CTAN sync 202009260303
Diffstat (limited to 'macros/latex/contrib/fontsetup/doc/fspsample.tex')
-rw-r--r-- | macros/latex/contrib/fontsetup/doc/fspsample.tex | 75 |
1 files changed, 0 insertions, 75 deletions
diff --git a/macros/latex/contrib/fontsetup/doc/fspsample.tex b/macros/latex/contrib/fontsetup/doc/fspsample.tex deleted file mode 100644 index 23ec736e02..0000000000 --- a/macros/latex/contrib/fontsetup/doc/fspsample.tex +++ /dev/null @@ -1,75 +0,0 @@ -%% 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} |