From f2d633715a7c607d03a75013e07d76446b4c1a25 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sat, 3 Oct 2026 16:10:45 -0400 Subject: [PATCH] On that convolution race with Henry. --- src/measure/measure/lebesgue-stieltjes.tex | 14 ++++++++++++-- 1 file changed, 12 insertions(+), 2 deletions(-) diff --git a/src/measure/measure/lebesgue-stieltjes.tex b/src/measure/measure/lebesgue-stieltjes.tex index 3879a25..8501e5f 100644 --- a/src/measure/measure/lebesgue-stieltjes.tex +++ b/src/measure/measure/lebesgue-stieltjes.tex @@ -46,7 +46,17 @@ \item[(U)] For any function $G: \real \to \real$ satisfying (1), $F - G$ is constant. \end{enumerate} - Conversely, if $F: \real \to \real$ is a Stieltjes function, then there exists a unique Borel measure $\mu_F: \cb_\real \to [0, \infty]$ satisfying (1), and $\mu_F$ is the \textbf{Lebesgue-Stieltjes measure} associated with $F$. + Conversely, if $F: \real \to \real$ is a Stieltjes function, then + \begin{enumerate} + \item[(1C)] There exists a unique Borel measure $\mu_F: \cb_\real \to [0, \infty]$ such that for all $-\infty < a < b < \infty$, + \[ + \mu_F((a, b]) = F(b) - F(a) + \] + \end{enumerate} + + In which case, $\mu_F$ is the \textbf{Lebesgue-Stieltjes measure} associated with $F$. + + \end{definition} \begin{proof}[Proof {{\cite[Theorem 1.16]{Folland}}}. ] (1): Let @@ -71,7 +81,7 @@ (U): If $G: \real \to \real$ satisfies (1), then $F(x) = G(x) - G(0)$. Hence $F - G$ is constant. - Converse: Let $\ci = \bracs{(a, b]| -\infty < a \le b < \infty}$ be the elementary family of h-intervals and + (1C): Let $\ci = \bracs{(a, b]| -\infty < a \le b < \infty}$ be the elementary family of h-intervals and \[ \alg = \bracs{\bigsqcup_{j = 1}^n (a_j, b_j] \bigg | \seqf{(a_j, b_j]} \subset \ci, n \in \natp} \]