Cleanup

src/fa/rs/bv.tex

@@ -8,10 +8,12 @@
     \[
     V_{\rho, p}(f) = \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1}))
     \]
+
     is the \textbf{variation} of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
     \[
     [f]_{\var, \rho} = \sup_{P \in \scp([a, b])}V_{\rho, P}(f)
     \]
+
     is the \textbf{total variation} of $f$ on $[a, b]$ with respect to $\rho$.
 
     If $E$ is a normed space, then the variation and total variation of $f$ is taken with respect to its norm.
@@ -45,12 +47,14 @@
     = \lim_{g, \fF}\sum_{j = 1}^n \rho(g(x_j) - g(x_{j - 1}))
     = \lim_{g \in \fF}V_{\rho, P}(g)
     \]
+
     By assumption (b), $[0, M_\rho]$ is in the filter generated by $V_{\rho, P}(\fF)$. Thus $V_{\rho, P}(f) \le M_\rho$. As this holds for all $P \in \scp([a, b])$, $V_{\rho, P}(f) \le M_\rho$, and $f \in BV([a, b]; E)$.
 
     (5): For each $n \in \nat^+$, let
     \[
     D_n = \bracs{x \in [a, b]|\forall \eps > 0, \exists y \in (x - \eps, x + \eps): \norm{f(x) - f(y)}_E \ge 1/n}
     \]
+
     then $D = \bigcup_{n \in \nat^+}D_n$ is the set of discontinuity points of $f$. If $D$ is uncountable, then there exists $N \in \nat^+$ such that $D_n$ is infinite.
 
     Fix $N \in \nat^+$. Let $E_1 = D_n \cap (a, b)$ and $I_1 = (a, b)$, then