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dominated-convergence.tex
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dominated-convergence.tex
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\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{relsize}
\usepackage[margin=1in, paperwidth=8.5in, paperheight=11in]{geometry}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{parskip}
\usepackage{microtype}
\pagenumbering{gobble}
\renewcommand{\thesection}{\Roman{section}}
\renewcommand{\thesubsection}{\thesection.\Roman{subsection}}
\theoremstyle{definition}
\newenvironment{example}[1]{\par\noindent\textit{Example.}\space#1}{}
\newenvironment{lemma}[1]{\par\noindent\textit{Lemma.}\space#1}{}
\newtheorem*{theorem}{Theorem}
\newtheorem*{claim}{Claim}
\newtheorem{definition}{Definition}
\newtheorem*{remark}{Remark}
\allowdisplaybreaks
\begin{document}
\begin{center}
\large \textbf{lebesgue's dominated convergence theorem} \\
{surely Clueless it is trivial measure theory} \\
{\small court. dashiel \\}
\end{center}
\begin{theorem}[Dominated Convergence Theorem]
Suppose $f_n$ is a sequence of measurable functions that converges pointwise to $f$ almost everywhere and $|f_n| \leq g$ with $\int g < \infty$. Then we have
\begin{equation*}
\lim_{n} \int f_n = \int f
\end{equation*}
\end{theorem}
\begin{proof}
Since $|f_n| \leq g < \infty$, it follows that $f_n < \infty$
\begin{align*}
|f_n| \leq g \implies \int |f_n| & \leq \int g < \infty
\intertext{By the monotonicity of the integral, since $|f| \leq g$, it follows that$ \int f < \infty$.}
|f_n - f| \leq |f_n| + |f| \leq 2g
\intertext{Define a measurable function $h_n = 2g - |f_n - f| \leq 0$. We can apply Fatou's lemma.}
\int \lim_{n} \inf h_n & \leq \lim_{n} \int \inf h_n \\
\int 2g & \leq \int 2g - \lim_{n} \sup |f_n - f| \\
0 \leq \lim_{n} \inf \int |f_n - f| & \leq \lim_{n} \sup \int |f_n - f| \leq 0 \\
\implies \lim_{n} \inf \int |f_n - f| & = \lim_{n} \sup \int |f_n - f| = 0
\intertext{Thus, the limit exists and}
\lim_{n} \int |f_n - f| = 0 \\
0 \leq \lim_{n} \left|\int f_n - \int f\right| & = \lim_{n} \left|\int f_n - f\right| \leq \lim_{n} \int|f_n - f| = 0 \\
\implies \lim_{n} \int f_n = \int f
\end{align*}
\end{proof}
\begin{remark}
oh wow i can't believe that i understood a measure theory proof
\end{remark}
\end{document}