Probability with Measure

\(\newcommand{\footnotename}{footnote}\) \(\def \LWRfootnote {1}\) \(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\) \(\let \LWRorighspace \hspace \) \(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\) \(\newcommand {\mathnormal }[1]{{#1}}\) \(\newcommand \ensuremath [1]{#1}\) \(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \) \(\newcommand {\setlength }[2]{}\) \(\newcommand {\addtolength }[2]{}\) \(\newcommand {\setcounter }[2]{}\) \(\newcommand {\addtocounter }[2]{}\) \(\newcommand {\arabic }[1]{}\) \(\newcommand {\number }[1]{}\) \(\newcommand {\noalign }[1]{\text {#1}\notag \\}\) \(\newcommand {\cline }[1]{}\) \(\newcommand {\directlua }[1]{\text {(directlua)}}\) \(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\) \(\newcommand {\protect }{}\) \(\def \LWRabsorbnumber #1 {}\) \(\def \LWRabsorbquotenumber "#1 {}\) \(\newcommand {\LWRabsorboption }[1][]{}\) \(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\) \(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\) \(\def \mathcode #1={\mathchar }\) \(\let \delcode \mathcode \) \(\let \delimiter \mathchar \) \(\def \oe {\unicode {x0153}}\) \(\def \OE {\unicode {x0152}}\) \(\def \ae {\unicode {x00E6}}\) \(\def \AE {\unicode {x00C6}}\) \(\def \aa {\unicode {x00E5}}\) \(\def \AA {\unicode {x00C5}}\) \(\def \o {\unicode {x00F8}}\) \(\def \O {\unicode {x00D8}}\) \(\def \l {\unicode {x0142}}\) \(\def \L {\unicode {x0141}}\) \(\def \ss {\unicode {x00DF}}\) \(\def \SS {\unicode {x1E9E}}\) \(\def \dag {\unicode {x2020}}\) \(\def \ddag {\unicode {x2021}}\) \(\def \P {\unicode {x00B6}}\) \(\def \copyright {\unicode {x00A9}}\) \(\def \pounds {\unicode {x00A3}}\) \(\let \LWRref \ref \) \(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\) \( \newcommand {\multicolumn }[3]{#3}\) \(\require {textcomp}\) \(\newcommand {\intertext }[1]{\text {#1}\notag \\}\) \(\let \Hat \hat \) \(\let \Check \check \) \(\let \Tilde \tilde \) \(\let \Acute \acute \) \(\let \Grave \grave \) \(\let \Dot \dot \) \(\let \Ddot \ddot \) \(\let \Breve \breve \) \(\let \Bar \bar \) \(\let \Vec \vec \) \(\DeclareMathOperator {\var }{var}\) \(\DeclareMathOperator {\cov }{cov}\) \(\newcommand {\nN }{n \in \mathbb {N}}\) \(\newcommand {\Br }{{\cal B}(\R )}\) \(\newcommand {\F }{{\cal F}}\) \(\newcommand {\ds }{\displaystyle }\) \(\newcommand {\st }{\stackrel {d}{=}}\) \(\newcommand {\uc }{\stackrel {uc}{\rightarrow }}\) \(\newcommand {\la }{\langle }\) \(\newcommand {\ra }{\rangle }\) \(\newcommand {\li }{\liminf _{n \rightarrow \infty }}\) \(\newcommand {\ls }{\limsup _{n \rightarrow \infty }}\) \(\newcommand {\limn }{\lim _{n \rightarrow \infty }}\) \(\def \to {\rightarrow }\) \(\def \iff {\Leftrightarrow }\) \(\def \sw {\subseteq }\) \(\def \mc {\mathcal }\) \(\def \mb {\mathbb }\) \(\def \sc {\setminus }\) \(\def \v {\textbf }\) \(\def \E {\mb {E}}\) \(\def \P {\mb {P}}\) \(\def \R {\mb {R}}\) \(\def \C {\mb {C}}\) \(\def \N {\mb {N}}\) \(\def \Q {\mb {Q}}\) \(\def \Z {\mb {Z}}\) \(\def \B {\mb {B}}\) \(\def \~{\sim }\) \(\def \-{\,;\,}\) \(\def \qed {$\blacksquare $}\) \(\def \1{\unicode {x1D7D9}}\) \(\def \cadlag {c\`{a}dl\`{a}g}\) \(\def \p {\partial }\) \(\def \l {\left }\) \(\def \r {\right }\) \(\def \Om {\Omega }\) \(\def \om {\omega }\)