From 36252d8fe0ee0ebac924b05a37d50481d84815e9 Mon Sep 17 00:00:00 2001 From: flavis Date: Sat, 7 Nov 2020 17:24:41 +0100 Subject: [PATCH] add proper cupdots --- lecture.cls | 16 ++++++++++++++++ ws2020/ana/uebungen/ana1.pdf | Bin 145929 -> 145901 bytes ws2020/ana/uebungen/ana1.tex | 11 ++++++----- 3 files changed, 22 insertions(+), 5 deletions(-) diff --git a/lecture.cls b/lecture.cls index 45c9d68..904ada4 100644 --- a/lecture.cls +++ b/lecture.cls @@ -162,6 +162,22 @@ % contradiction \newcommand{\contr}{\text{\Large\lightning}} +% disjoint unions: provides cupdot and bigcupdot +\makeatletter +\def\moverlay{\mathpalette\mov@rlay} +\def\mov@rlay#1#2{\leavevmode\vtop{% + \baselineskip\z@skip \lineskiplimit-\maxdimen + \ialign{\hfil$\m@th#1##$\hfil\cr#2\crcr}}} +\newcommand{\charfusion}[3][\mathord]{ + #1{\ifx#1\mathop\vphantom{#2}\fi + \mathpalette\mov@rlay{#2\cr#3} + } + \ifx#1\mathop\expandafter\displaylimits\fi} +\makeatother + +\newcommand{\cupdot}{\charfusion[\mathbin]{\cup}{\cdot}} +\newcommand{\bigcupdot}{\charfusion[\mathop]{\bigcup}{\cdot}} + \ExplSyntaxOn % S-tackrelcompatible ALIGN environment diff --git a/ws2020/ana/uebungen/ana1.pdf b/ws2020/ana/uebungen/ana1.pdf index f0a1f58b2e28d3ec856664b198e87e7f0b2c767e..7952bfbc95135f523335fd41cb83c94b7ec6d8bd 100644 GIT binary patch delta 31718 zcmV(>K-j;D^a$|uxF#}vITpTNTu_1Ce%U!>_48D5x_rI~p-TH)1?k05d zr<03UCtv-Msfi;_b3J)=JCU3yKCw&?X3XT(&E)m;`Es`SxSp?1Ga&8w4to-o)Wov><8)DhwM!0c52EO`sLg6Dj`49#^Or(Gz z1|AoaA5OmCo-#SDaM5YCArhW{j8TLOSseEw|B%nuBy6MlWrIk<5T_J;6MpbRKKFxR z>Ua#(gi2G5xhzH`voHpat{Rcgm*p7NIm;Cl44DYVi0zo{hmY$`{y++nUe1;!aqs4f zO>(A5W*O0fPcjBmaaAp$=p6oG&GRR_QbfA#`F||8B-685Dyk72p^qbflDhHaxnImy zrPSQakm-abmaFmeU4C5N&zI|xRw-Q=r63Ll9u#phqRTL0Ea940YdpsYpNju)uY{4S;`tLej1Qi) z6`fz^%ex%4juZ}7o0;cdgU``sdV%bpOJZu1eMLD+NpLjM_6k;iLdnXewyl8UdcX=u zx0O%qXqHgAUFQO^0-V@d!k%HfNDkAvi`q~yA(3eu(b_gPl1MV_R2j@pB(MSoxx`M? 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Damit folgt wegen $\mathcal{D}$ Dynkinsystem \[ - \bigcup_{i \in \N} A_i = \mathop{\dot{\bigcup_{i \in \N}}} B_i \in \mathscr{D} + \bigcup_{i \in \N} A_i = \bigcupdot_{i \in \N} B_i \in \mathscr{D} .\] \end{enumerate} \end{proof} @@ -191,16 +191,17 @@ \item Sei $A \in \mathscr{H}(D)$. Dann ist $A \cap D \in \mathscr{D}_0$. Da $\mathscr{D}_0$ Dynkinsystem folgt: \begin{align*} - A^{c} \cap D = (X \setminus A) \cap D = (X \cap D) \setminus (A \cap D) - = \left( (X \cap D)^{c} \mathop{\dot{\cup}} (A \cap D) \right)^{c} + A^{c} \cap D + = D \setminus (A \cap D) + = \left( D^{c} \cupdot (A \cap D) \right)^{c} \in \mathscr{D}_0 .\end{align*} \item Sei $A_i \in \mathscr{H}(D)$ $\forall i \in \N$ mit $A_i \cap A_j = \emptyset$ $\forall i, j \in \N, i \neq j$. Dann folgt direkt, da die $A_i$ paarweise disjunkt sind und $\mathscr{D}_0$ Dynkinsystem: \[ - \left( \bigcup_{i \in \N} A_i \right) \cap D - = \mathop{\dot{\bigcup_{i \in \N}}} (\underbrace{A_i \cap D}_{ \in \mathscr{D}_0}) + \left( \bigcupdot_{i \in \N} A_i \right) \cap D + = \bigcupdot_{i \in \N} (\underbrace{A_i \cap D}_{ \in \mathscr{D}_0}) \in \mathscr{D}_0 .\] \end{enumerate}