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\begin{document}
\section{Lebesgue-Stieltjes Integral}

\begin{defi}
    Let $F: \realn \rightarrow \realn$ be a monotonically increasing, continuous function. Then we set 
    \begin{align*}
        \lambda_F(\varnothing) := 0 && \lambda_F((a, b]) = F(b) - F(a), ~~(a, b] \in \intervals
    \end{align*}
\end{defi}

\begin{thm}
    $\lambda_F$ is a measure on $H$.
\end{thm}
\begin{proof}
    Without proof.
\end{proof}

\begin{defi}
    The integral 
    \[
        \int_A f \dd{\lambda_F}
    \]
    is called the Lebesgue-Stieltjes integral on $\realn$ and is denoted by 
    \[
        \int_A f(x) \dd{F(x)} := \int_A f \dd{\lambda_F}
    \]
    If $A = [a, b]$, then we write
    \[
        \int_a^b f(x) \dd{F(x)}
    \]
\end{defi}
\end{document}