Probability with Measure
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\DeclareMathOperator {\var }{var}\)
\(\DeclareMathOperator {\cov }{cov}\)
\(\newcommand {\nN }{n \in \mathbb {N}}\)
\(\newcommand {\Br }{{\cal B}(\R )}\)
\(\newcommand {\F }{{\cal F}}\)
\(\newcommand {\ds }{\displaystyle }\)
\(\newcommand {\st }{\stackrel {d}{=}}\)
\(\newcommand {\uc }{\stackrel {uc}{\rightarrow }}\)
\(\newcommand {\la }{\langle }\)
\(\newcommand {\ra }{\rangle }\)
\(\newcommand {\li }{\liminf _{n \rightarrow \infty }}\)
\(\newcommand {\ls }{\limsup _{n \rightarrow \infty }}\)
\(\newcommand {\limn }{\lim _{n \rightarrow \infty }}\)
\(\def \to {\rightarrow }\)
\(\def \iff {\Leftrightarrow }\)
\(\def \ra {\Rightarrow }\)
\(\def \sw {\subseteq }\)
\(\def \mc {\mathcal }\)
\(\def \mb {\mathbb }\)
\(\def \sc {\setminus }\)
\(\def \v {\textbf }\)
\(\def \E {\mb {E}}\)
\(\def \P {\mb {P}}\)
\(\def \R {\mb {R}}\)
\(\def \C {\mb {C}}\)
\(\def \N {\mb {N}}\)
\(\def \Q {\mb {Q}}\)
\(\def \Z {\mb {Z}}\)
\(\def \B {\mb {B}}\)
\(\def \~{\sim }\)
\(\def \-{\,;\,}\)
\(\def \qed {$\blacksquare $}\)
\(\def \1{\unicode {x1D7D9}}\)
\(\def \cadlag {c\`{a}dl\`{a}g}\)
\(\def \p {\partial }\)
\(\def \l {\left }\)
\(\def \r {\right }\)
\(\def \Om {\Omega }\)
\(\def \om {\omega }\)
\(\def \eps {\epsilon }\)
\(\def \de {\delta }\)
\(\def \ov {\overline }\)
\(\def \sr {\stackrel }\)
\(\def \Lp {\mc {L}^p}\)
\(\def \Lq {\mc {L}^q}\)
\(\def \Lone {\mc {L}^1}\)
\(\def \Ltwo {\mc {L}^2}\)
\(\def \toae {\sr {\rm a.e.}{\to }}\)
\(\def \toas {\sr {\rm a.s.}{\to }}\)
\(\def \top {\sr {\mb {\P }}{\to }}\)
\(\def \tod {\sr {\rm d}{\to }}\)
\(\def \toLp {\sr {\Lp }{\to }}\)
\(\def \toLq {\sr {\Lq }{\to }}\)
\(\def \eqae {\sr {\rm a.e.}{=}}\)
\(\def \eqas {\sr {\rm a.s.}{=}}\)
\(\def \eqd {\sr {\rm d}{=}}\)
\(\def \Sa {(S1)}\)
\(\def \Sb {(S2)}\)
\(\def \Sc {(S3)}\)
\(\def \Scp {(S3')}\)
\(\def \Ma {(M1)}\)
\(\def \Mb {(M2)}\)
\(\def \La {(L1)}\)
\(\def \Lb {(L2)}\)
\(\def \Lc {(L3)}\)
\(\def \Ld {(L4)}\)
\(\def \Le {(L5)}\)
Chapter 0 Introduction
0.1 Organization
0.1.1 Syllabus
These notes are for two courses: MAS31002 and MAS61022.
Some sections of the course are included in MAS61022 but not in MAS31002. These sections are marked with a \((\Delta )\) symbol. We will not cover these sections in lectures. Students taking MAS61022
should study these sections independently.
Some parts of the notes are marked with a \((\star )\) symbol, which means they are off-syllabus. These are often cases where detailed connections can be made to and from other parts of mathematics.
0.1.2 Problem sheets
The exercises are divided up according to the chapters of the course. Some exercises are marked as ‘challenge questions’ – these are intended to offer a serious, time consuming challenge to the best students.
Aside from challenge questions, it is expected that students will attempt all exercises (for the version of the course they are taking) and review their own solutions using the typed solutions provided in the online
version of these notes, in Appendix B.
At three points during each semester, an assignment of additional exercises will be set. About one week later, a mark scheme will be posted, and you should self-mark your solutions.
0.1.3 Examination
The course will be examined in the summer sitting. Parts of the course marked with a \((\Delta )\) are examinable for MAS61022 but not for MAS31002. Parts of the course marked with a \((\star )\) will not
be examined (for everyone). Some advice on how to structure your revision can be found in Appendix A.
0.1.4 Website
Further information, including the timetable, can be found on