35 lines
828 B
TeX
35 lines
828 B
TeX
% !TeX root = ../../script.tex
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\documentclass[../../script.tex]{subfiles}
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\begin{document}
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\section{Lebesgue-Stieltjes Integral}
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\begin{defi}
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Let $F: \realn \rightarrow \realn$ be a monotonically increasing, continuous function. Then we set
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\begin{align*}
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\lambda_F(\varnothing) := 0 && \lambda_F((a, b]) = F(b) - F(a), ~~(a, b] \in \intervals
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\end{align*}
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\end{defi}
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\begin{thm}
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$\lambda_F$ is a measure on $H$.
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\end{thm}
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\begin{proof}
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Without proof.
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\end{proof}
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\begin{defi}
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The integral
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\[
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\int_A f \dd{\lambda_F}
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\]
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is called the Lebesgue-Stieltjes integral on $\realn$ and is denoted by
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\[
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\int_A f(x) \dd{F(x)} := \int_A f \dd{\lambda_F}
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\]
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If $A = [a, b]$, then we write
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\[
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\int_a^b f(x) \dd{F(x)}
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\]
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\end{defi}
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\end{document} |