Sat Dec 23 20:35:26 GMT 2017

III_M/advanced_probability.tex

@@ -1155,15 +1155,15 @@ \subsection{Applications of martingales}
 
 \begin{thm}[Radon--Nikodym]\index{Radon--Nikodym theorem}
   Let $(\Omega, \mathcal{F})$ be a measurable space, and $\Q$ and $\P$ be two probability measures on $(\Omega, \mathcal{F})$. Then the following are equivalent:
-  \begin{itemize}
+  \begin{enumerate}
     \item $\Q$ is absolutely continuous with respect to $\P$, i.e.\ for any $A \in \mathcal{F}$, if $\P(A) = 0$, then $\Q(A) = 0$.
     \item For any $\varepsilon > 0$, there exists $\delta > 0$ such that for all $A \in \mathcal{F}$, if $\P(A) \leq \delta$, then $\Q(A) \leq \varepsilon$.
     \item There exists a random variable $X \geq 0$ such that
       \[
         \Q(A) = \E_{\P}(X \mathbf{1}_A).
       \]
       In this case, $X$ is called the \term{Radon--Nikodym derivative} of $\Q$ with respect to $\P$, and we write $X = \frac{\d \Q}{\d \P}$.
-  \end{itemize}
+  \end{enumerate}
 \end{thm}
 Note that this theorem works for all finite measures by scaling, and thus for $\sigma$-finite measures by partitioning $\Omega$ into sets of finite measure.
 

III_M/algebraic_topology_iii.tex

@@ -3279,7 +3279,7 @@ \subsection{Gysin sequence}
 \[
   \begin{tikzcd}
     H^{i + d}(E, E^\#) \ar[r] & H^{i + d}(E) \ar[d, "s_0^*", xshift=2] \ar[r] & H^{i + d} (E^\#) \ar[r] \ar[d, "j^*"] & H^{i + d + 1}(E, E^\#)\\
-    H^i(X) \ar[u, "\Phi"] \ar[r, "\ph \smile e(E)"] & H^{i + d}(X) \ar[u, xshift=-2, "\pi^*"] \ar[r, "p"] & H^{i + d}(S(E)) \ar[r, "p_!"] & H^{i + 1}(X)\ar[u, "\Phi"]
+    H^i(X) \ar[u, "\Phi"] \ar[r, "\ph \smile e(E)"] & H^{i + d}(X) \ar[u, xshift=-2, "\pi^*"] \ar[r, "p^*"] & H^{i + d}(S(E)) \ar[r, "p_!"] & H^{i + 1}(X)\ar[u, "\Phi"]
   \end{tikzcd}
 \]
 where $p: S(E) \to E$ is the projection, and $p_!$ is whatever makes the diagram commutes (since $j^*$ and $\Phi$ are isomorphisms). The bottom sequence is the \term{Gysin sequence}, and it is exact because the top row is exact. This is in fact a long exact sequence of $H^*(X; R)$-modules, i.e.\ the maps commute with cup products.

III_M/combinatorics.tex

@@ -1378,7 +1378,7 @@ \section{Projections}
     &\leq \int \prod_{A \in A_-} |K(x)_A|^{1/k} \prod_{A \in \mathcal{A}_+} |K(x)_A|^{1/k}\;\d x\tag{by induction}\\
     &\leq \prod_{A \in \mathcal{A}_-} |K_A|^{1/k} \int \prod_{A \in \mathcal{A}_+}|K(x)_A|^{1/k}\;\d x\tag{by (1)}\\
     &\leq \prod_{A \leq |\mathcal{A}_-} |K_A|^{1/k} \prod_{A \in \mathcal{A}_+} \left(\int |K(x)_A|\right)^{1/k} \tag{by H\"older}\\
-    &= \prod_{A \in \mathcal{A}} |K_A|^{1/k} \prod_{A \in \mathcal{A}_+} |K_{A \cup \{n\}}|^{1/k}\tag{by (2)}.\qedhere
+    &= \prod_{A \in \mathcal{A}} |K_A|^{1/k} \prod_{A \in \mathcal{A}_+} |K_{A \cup \{n\}}|^{1/k}\tag{by (2)}.%\qedhere
   \end{align*}
 \end{proof}
 This theorem is great, but we can do better. In fact,