Rewrite the Probability theory.
Introduced pinfreal as real numbers with infinity.
Use pinfreal as value for measures.
Introduces Lebesgue Measure based on the integral in Multivariate Analysis.
Proved Radon Nikodym for arbitrary sigma finite measure spaces.
%%
%% definitions of standard Isabelle symbols
%%
\newcommand{\isasymzero}{\isamath{\mathbf{0}}} %requires amssymb
\newcommand{\isasymone}{\isamath{\mathbf{1}}} %requires amssymb
\newcommand{\isasymtwo}{\isamath{\mathbf{2}}} %requires amssymb
\newcommand{\isasymthree}{\isamath{\mathbf{3}}} %requires amssymb
\newcommand{\isasymfour}{\isamath{\mathbf{4}}} %requires amssymb
\newcommand{\isasymfive}{\isamath{\mathbf{5}}} %requires amssymb
\newcommand{\isasymsix}{\isamath{\mathbf{6}}} %requires amssymb
\newcommand{\isasymseven}{\isamath{\mathbf{7}}} %requires amssymb
\newcommand{\isasymeight}{\isamath{\mathbf{8}}} %requires amssymb
\newcommand{\isasymnine}{\isamath{\mathbf{9}}} %requires amssymb
\newcommand{\isasymA}{\isamath{\mathcal{A}}}
\newcommand{\isasymB}{\isamath{\mathcal{B}}}
\newcommand{\isasymC}{\isamath{\mathcal{C}}}
\newcommand{\isasymD}{\isamath{\mathcal{D}}}
\newcommand{\isasymE}{\isamath{\mathcal{E}}}
\newcommand{\isasymF}{\isamath{\mathcal{F}}}
\newcommand{\isasymG}{\isamath{\mathcal{G}}}
\newcommand{\isasymH}{\isamath{\mathcal{H}}}
\newcommand{\isasymI}{\isamath{\mathcal{I}}}
\newcommand{\isasymJ}{\isamath{\mathcal{J}}}
\newcommand{\isasymK}{\isamath{\mathcal{K}}}
\newcommand{\isasymL}{\isamath{\mathcal{L}}}
\newcommand{\isasymM}{\isamath{\mathcal{M}}}
\newcommand{\isasymN}{\isamath{\mathcal{N}}}
\newcommand{\isasymO}{\isamath{\mathcal{O}}}
\newcommand{\isasymP}{\isamath{\mathcal{P}}}
\newcommand{\isasymQ}{\isamath{\mathcal{Q}}}
\newcommand{\isasymR}{\isamath{\mathcal{R}}}
\newcommand{\isasymS}{\isamath{\mathcal{S}}}
\newcommand{\isasymT}{\isamath{\mathcal{T}}}
\newcommand{\isasymU}{\isamath{\mathcal{U}}}
\newcommand{\isasymV}{\isamath{\mathcal{V}}}
\newcommand{\isasymW}{\isamath{\mathcal{W}}}
\newcommand{\isasymX}{\isamath{\mathcal{X}}}
\newcommand{\isasymY}{\isamath{\mathcal{Y}}}
\newcommand{\isasymZ}{\isamath{\mathcal{Z}}}
\newcommand{\isasyma}{\isamath{\mathrm{a}}}
\newcommand{\isasymb}{\isamath{\mathrm{b}}}
\newcommand{\isasymc}{\isamath{\mathrm{c}}}
\newcommand{\isasymd}{\isamath{\mathrm{d}}}
\newcommand{\isasyme}{\isamath{\mathrm{e}}}
\newcommand{\isasymf}{\isamath{\mathrm{f}}}
\newcommand{\isasymg}{\isamath{\mathrm{g}}}
\newcommand{\isasymh}{\isamath{\mathrm{h}}}
\newcommand{\isasymi}{\isamath{\mathrm{i}}}
\newcommand{\isasymj}{\isamath{\mathrm{j}}}
\newcommand{\isasymk}{\isamath{\mathrm{k}}}
\newcommand{\isasyml}{\isamath{\mathrm{l}}}
\newcommand{\isasymm}{\isamath{\mathrm{m}}}
\newcommand{\isasymn}{\isamath{\mathrm{n}}}
\newcommand{\isasymo}{\isamath{\mathrm{o}}}
\newcommand{\isasymp}{\isamath{\mathrm{p}}}
\newcommand{\isasymq}{\isamath{\mathrm{q}}}
\newcommand{\isasymr}{\isamath{\mathrm{r}}}
\newcommand{\isasyms}{\isamath{\mathrm{s}}}
\newcommand{\isasymt}{\isamath{\mathrm{t}}}
\newcommand{\isasymu}{\isamath{\mathrm{u}}}
\newcommand{\isasymv}{\isamath{\mathrm{v}}}
\newcommand{\isasymw}{\isamath{\mathrm{w}}}
\newcommand{\isasymx}{\isamath{\mathrm{x}}}
\newcommand{\isasymy}{\isamath{\mathrm{y}}}
\newcommand{\isasymz}{\isamath{\mathrm{z}}}
\newcommand{\isasymAA}{\isamath{\mathfrak{A}}} %requires eufrak
\newcommand{\isasymBB}{\isamath{\mathfrak{B}}} %requires eufrak
\newcommand{\isasymCC}{\isamath{\mathfrak{C}}} %requires eufrak
\newcommand{\isasymDD}{\isamath{\mathfrak{D}}} %requires eufrak
\newcommand{\isasymEE}{\isamath{\mathfrak{E}}} %requires eufrak
\newcommand{\isasymFF}{\isamath{\mathfrak{F}}} %requires eufrak
\newcommand{\isasymGG}{\isamath{\mathfrak{G}}} %requires eufrak
\newcommand{\isasymHH}{\isamath{\mathfrak{H}}} %requires eufrak
\newcommand{\isasymII}{\isamath{\mathfrak{I}}} %requires eufrak
\newcommand{\isasymJJ}{\isamath{\mathfrak{J}}} %requires eufrak
\newcommand{\isasymKK}{\isamath{\mathfrak{K}}} %requires eufrak
\newcommand{\isasymLL}{\isamath{\mathfrak{L}}} %requires eufrak
\newcommand{\isasymMM}{\isamath{\mathfrak{M}}} %requires eufrak
\newcommand{\isasymNN}{\isamath{\mathfrak{N}}} %requires eufrak
\newcommand{\isasymOO}{\isamath{\mathfrak{O}}} %requires eufrak
\newcommand{\isasymPP}{\isamath{\mathfrak{P}}} %requires eufrak
\newcommand{\isasymQQ}{\isamath{\mathfrak{Q}}} %requires eufrak
\newcommand{\isasymRR}{\isamath{\mathfrak{R}}} %requires eufrak
\newcommand{\isasymSS}{\isamath{\mathfrak{S}}} %requires eufrak
\newcommand{\isasymTT}{\isamath{\mathfrak{T}}} %requires eufrak
\newcommand{\isasymUU}{\isamath{\mathfrak{U}}} %requires eufrak
\newcommand{\isasymVV}{\isamath{\mathfrak{V}}} %requires eufrak
\newcommand{\isasymWW}{\isamath{\mathfrak{W}}} %requires eufrak
\newcommand{\isasymXX}{\isamath{\mathfrak{X}}} %requires eufrak
\newcommand{\isasymYY}{\isamath{\mathfrak{Y}}} %requires eufrak
\newcommand{\isasymZZ}{\isamath{\mathfrak{Z}}} %requires eufrak
\newcommand{\isasymaa}{\isamath{\mathfrak{a}}} %requires eufrak
\newcommand{\isasymbb}{\isamath{\mathfrak{b}}} %requires eufrak
\newcommand{\isasymcc}{\isamath{\mathfrak{c}}} %requires eufrak
\newcommand{\isasymdd}{\isamath{\mathfrak{d}}} %requires eufrak
\newcommand{\isasymee}{\isamath{\mathfrak{e}}} %requires eufrak
\newcommand{\isasymff}{\isamath{\mathfrak{f}}} %requires eufrak
\newcommand{\isasymgg}{\isamath{\mathfrak{g}}} %requires eufrak
\newcommand{\isasymhh}{\isamath{\mathfrak{h}}} %requires eufrak
\newcommand{\isasymii}{\isamath{\mathfrak{i}}} %requires eufrak
\newcommand{\isasymjj}{\isamath{\mathfrak{j}}} %requires eufrak
\newcommand{\isasymkk}{\isamath{\mathfrak{k}}} %requires eufrak
\newcommand{\isasymll}{\isamath{\mathfrak{l}}} %requires eufrak
\newcommand{\isasymmm}{\isamath{\mathfrak{m}}} %requires eufrak
\newcommand{\isasymnn}{\isamath{\mathfrak{n}}} %requires eufrak
\newcommand{\isasymoo}{\isamath{\mathfrak{o}}} %requires eufrak
\newcommand{\isasympp}{\isamath{\mathfrak{p}}} %requires eufrak
\newcommand{\isasymqq}{\isamath{\mathfrak{q}}} %requires eufrak
\newcommand{\isasymrr}{\isamath{\mathfrak{r}}} %requires eufrak
\newcommand{\isasymss}{\isamath{\mathfrak{s}}} %requires eufrak
\newcommand{\isasymtt}{\isamath{\mathfrak{t}}} %requires eufrak
\newcommand{\isasymuu}{\isamath{\mathfrak{u}}} %requires eufrak
\newcommand{\isasymvv}{\isamath{\mathfrak{v}}} %requires eufrak
\newcommand{\isasymww}{\isamath{\mathfrak{w}}} %requires eufrak
\newcommand{\isasymxx}{\isamath{\mathfrak{x}}} %requires eufrak
\newcommand{\isasymyy}{\isamath{\mathfrak{y}}} %requires eufrak
\newcommand{\isasymzz}{\isamath{\mathfrak{z}}} %requires eufrak
\newcommand{\isasymalpha}{\isamath{\alpha}}
\newcommand{\isasymbeta}{\isamath{\beta}}
\newcommand{\isasymgamma}{\isamath{\gamma}}
\newcommand{\isasymdelta}{\isamath{\delta}}
\newcommand{\isasymepsilon}{\isamath{\varepsilon}}
\newcommand{\isasymzeta}{\isamath{\zeta}}
\newcommand{\isasymeta}{\isamath{\eta}}
\newcommand{\isasymtheta}{\isamath{\vartheta}}
\newcommand{\isasymiota}{\isamath{\iota}}
\newcommand{\isasymkappa}{\isamath{\kappa}}
\newcommand{\isasymlambda}{\isamath{\lambda}}
\newcommand{\isasymmu}{\isamath{\mu}}
\newcommand{\isasymnu}{\isamath{\nu}}
\newcommand{\isasymxi}{\isamath{\xi}}
\newcommand{\isasympi}{\isamath{\pi}}
\newcommand{\isasymrho}{\isamath{\varrho}}
\newcommand{\isasymsigma}{\isamath{\sigma}}
\newcommand{\isasymtau}{\isamath{\tau}}
\newcommand{\isasymupsilon}{\isamath{\upsilon}}
\newcommand{\isasymphi}{\isamath{\varphi}}
\newcommand{\isasymchi}{\isamath{\chi}}
\newcommand{\isasympsi}{\isamath{\psi}}
\newcommand{\isasymomega}{\isamath{\omega}}
\newcommand{\isasymGamma}{\isamath{\Gamma}}
\newcommand{\isasymDelta}{\isamath{\Delta}}
\newcommand{\isasymTheta}{\isamath{\Theta}}
\newcommand{\isasymLambda}{\isamath{\Lambda}}
\newcommand{\isasymXi}{\isamath{\Xi}}
\newcommand{\isasymPi}{\isamath{\Pi}}
\newcommand{\isasymSigma}{\isamath{\Sigma}}
\newcommand{\isasymUpsilon}{\isamath{\Upsilon}}
\newcommand{\isasymPhi}{\isamath{\Phi}}
\newcommand{\isasymPsi}{\isamath{\Psi}}
\newcommand{\isasymOmega}{\isamath{\Omega}}
\newcommand{\isasymbool}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{B}}}
\newcommand{\isasymcomplex}{\isamath{\mathrm{C}\mkern-15mu{\phantom{\mathrm{t}}\vrule}\mkern9mu}}
\newcommand{\isasymnat}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{N}}}
\newcommand{\isasymrat}{\isamath{\mathrm{Q}\mkern-16mu{\phantom{\mathrm{t}}\vrule}\mkern10mu}}
\newcommand{\isasymreal}{\isamath{\mathrm{I}\mkern-3.8mu\mathrm{R}}}
\newcommand{\isasymint}{\isamath{\mathsf{Z}\mkern-7.5mu\mathsf{Z}}}
\newcommand{\isasymleftarrow}{\isamath{\leftarrow}}
\newcommand{\isasymlongleftarrow}{\isamath{\longleftarrow}}
\newcommand{\isasymrightarrow}{\isamath{\rightarrow}}
\newcommand{\isasymlongrightarrow}{\isamath{\longrightarrow}}
\newcommand{\isasymLeftarrow}{\isamath{\Leftarrow}}
\newcommand{\isasymLongleftarrow}{\isamath{\Longleftarrow}}
\newcommand{\isasymRightarrow}{\isamath{\Rightarrow}}
\newcommand{\isasymLongrightarrow}{\isamath{\Longrightarrow}}
\newcommand{\isasymleftrightarrow}{\isamath{\leftrightarrow}}
\newcommand{\isasymlongleftrightarrow}{\isamath{\longleftrightarrow}}
\newcommand{\isasymLeftrightarrow}{\isamath{\Leftrightarrow}}
\newcommand{\isasymLongleftrightarrow}{\isamath{\Longleftrightarrow}}
\newcommand{\isasymmapsto}{\isamath{\mapsto}}
\newcommand{\isasymlongmapsto}{\isamath{\longmapsto}}
\newcommand{\isasymmidarrow}{\isamath{\relbar}}
\newcommand{\isasymMidarrow}{\isamath{\Relbar}}
\newcommand{\isasymhookleftarrow}{\isamath{\hookleftarrow}}
\newcommand{\isasymhookrightarrow}{\isamath{\hookrightarrow}}
\newcommand{\isasymleftharpoondown}{\isamath{\leftharpoondown}}
\newcommand{\isasymrightharpoondown}{\isamath{\rightharpoondown}}
\newcommand{\isasymleftharpoonup}{\isamath{\leftharpoonup}}
\newcommand{\isasymrightharpoonup}{\isamath{\rightharpoonup}}
\newcommand{\isasymrightleftharpoons}{\isamath{\rightleftharpoons}}
\newcommand{\isasymleadsto}{\isamath{\leadsto}} %requires amssymb
\newcommand{\isasymdownharpoonleft}{\isamath{\downharpoonleft}} %requires amssymb
\newcommand{\isasymdownharpoonright}{\isamath{\downharpoonright}} %requires amssymb
\newcommand{\isasymupharpoonleft}{\isamath{\upharpoonleft}} %requires amssymb
\newcommand{\isasymupharpoonright}{\isamath{\upharpoonright}} %requires amssymb
\newcommand{\isasymrestriction}{\isamath{\restriction}} %requires amssymb
\newcommand{\isasymColon}{\isamath{\mathrel{::}}}
\newcommand{\isasymup}{\isamath{\uparrow}}
\newcommand{\isasymUp}{\isamath{\Uparrow}}
\newcommand{\isasymdown}{\isamath{\downarrow}}
\newcommand{\isasymDown}{\isamath{\Downarrow}}
\newcommand{\isasymupdown}{\isamath{\updownarrow}}
\newcommand{\isasymUpdown}{\isamath{\Updownarrow}}
\newcommand{\isasymlangle}{\isamath{\langle}}
\newcommand{\isasymrangle}{\isamath{\rangle}}
\newcommand{\isasymlceil}{\isamath{\lceil}}
\newcommand{\isasymrceil}{\isamath{\rceil}}
\newcommand{\isasymlfloor}{\isamath{\lfloor}}
\newcommand{\isasymrfloor}{\isamath{\rfloor}}
\newcommand{\isasymlparr}{\isamath{\mathopen{(\mkern-3mu\mid}}}
\newcommand{\isasymrparr}{\isamath{\mathclose{\mid\mkern-3mu)}}}
\newcommand{\isasymlbrakk}{\isamath{\mathopen{\lbrack\mkern-3mu\lbrack}}}
\newcommand{\isasymrbrakk}{\isamath{\mathclose{\rbrack\mkern-3mu\rbrack}}}
\newcommand{\isasymlbrace}{\isamath{\mathopen{\lbrace\mkern-4.5mu\mid}}}
\newcommand{\isasymrbrace}{\isamath{\mathclose{\mid\mkern-4.5mu\rbrace}}}
\newcommand{\isasymguillemotleft}{\isatext{\flqq}} %requires babel
\newcommand{\isasymguillemotright}{\isatext{\frqq}} %requires babel
\newcommand{\isasymbottom}{\isamath{\bot}}
\newcommand{\isasymtop}{\isamath{\top}}
\newcommand{\isasymand}{\isamath{\wedge}}
\newcommand{\isasymAnd}{\isamath{\bigwedge}}
\newcommand{\isasymor}{\isamath{\vee}}
\newcommand{\isasymOr}{\isamath{\bigvee}}
\newcommand{\isasymforall}{\isamath{\forall\,}}
\newcommand{\isasymexists}{\isamath{\exists\,}}
\newcommand{\isasymnexists}{\isamath{\nexists\,}} %requires amssymb
\newcommand{\isasymnot}{\isamath{\neg}}
\newcommand{\isasymbox}{\isamath{\Box}} %requires amssymb
\newcommand{\isasymdiamond}{\isamath{\Diamond}} %requires amssymb
\newcommand{\isasymturnstile}{\isamath{\vdash}}
\newcommand{\isasymTurnstile}{\isamath{\models}}
\newcommand{\isasymtturnstile}{\isamath{\vdash\!\!\!\vdash}}
\newcommand{\isasymTTurnstile}{\isamath{\mid\!\models}}
\newcommand{\isasymstileturn}{\isamath{\dashv}}
\newcommand{\isasymsurd}{\isamath{\surd}}
\newcommand{\isasymle}{\isamath{\le}}
\newcommand{\isasymge}{\isamath{\ge}}
\newcommand{\isasymlless}{\isamath{\ll}}
\newcommand{\isasymggreater}{\isamath{\gg}}
\newcommand{\isasymlesssim}{\isamath{\lesssim}} %requires amssymb
\newcommand{\isasymgreatersim}{\isamath{\gtrsim}} %requires amssymb
\newcommand{\isasymlessapprox}{\isamath{\lessapprox}} %requires amssymb
\newcommand{\isasymgreaterapprox}{\isamath{\gtrapprox}} %requires amssymb
\newcommand{\isasymin}{\isamath{\in}}
\newcommand{\isasymnotin}{\isamath{\notin}}
\newcommand{\isasymsubset}{\isamath{\subset}}
\newcommand{\isasymsupset}{\isamath{\supset}}
\newcommand{\isasymsubseteq}{\isamath{\subseteq}}
\newcommand{\isasymsupseteq}{\isamath{\supseteq}}
\newcommand{\isasymsqsubset}{\isamath{\sqsubset}} %requires amssymb
\newcommand{\isasymsqsupset}{\isamath{\sqsupset}} %requires amssymb
\newcommand{\isasymsqsubseteq}{\isamath{\sqsubseteq}}
\newcommand{\isasymsqsupseteq}{\isamath{\sqsupseteq}}
\newcommand{\isasyminter}{\isamath{\cap}}
\newcommand{\isasymInter}{\isamath{\bigcap\,}}
\newcommand{\isasymunion}{\isamath{\cup}}
\newcommand{\isasymUnion}{\isamath{\bigcup\,}}
\newcommand{\isasymsqunion}{\isamath{\sqcup}}
\newcommand{\isasymSqunion}{\isamath{\bigsqcup\,}}
\newcommand{\isasymsqinter}{\isamath{\sqcap}}
\newcommand{\isasymSqinter}{\isamath{\bigsqcap\,}} %requires stmaryrd
\newcommand{\isasymsetminus}{\isamath{\setminus}}
\newcommand{\isasympropto}{\isamath{\propto}}
\newcommand{\isasymuplus}{\isamath{\uplus}}
\newcommand{\isasymUplus}{\isamath{\biguplus\,}}
\newcommand{\isasymnoteq}{\isamath{\not=}}
\newcommand{\isasymsim}{\isamath{\sim}}
\newcommand{\isasymdoteq}{\isamath{\doteq}}
\newcommand{\isasymsimeq}{\isamath{\simeq}}
\newcommand{\isasymapprox}{\isamath{\approx}}
\newcommand{\isasymasymp}{\isamath{\asymp}}
\newcommand{\isasymcong}{\isamath{\cong}}
\newcommand{\isasymsmile}{\isamath{\smile}}
\newcommand{\isasymequiv}{\isamath{\equiv}}
\newcommand{\isasymfrown}{\isamath{\frown}}
\newcommand{\isasymJoin}{\isamath{\Join}} %requires amssymb
\newcommand{\isasymbowtie}{\isamath{\bowtie}}
\newcommand{\isasymprec}{\isamath{\prec}}
\newcommand{\isasymsucc}{\isamath{\succ}}
\newcommand{\isasympreceq}{\isamath{\preceq}}
\newcommand{\isasymsucceq}{\isamath{\succeq}}
\newcommand{\isasymparallel}{\isamath{\parallel}}
\newcommand{\isasymbar}{\isamath{\mid}}
\newcommand{\isasymplusminus}{\isamath{\pm}}
\newcommand{\isasymminusplus}{\isamath{\mp}}
\newcommand{\isasymtimes}{\isamath{\times}}
\newcommand{\isasymdiv}{\isamath{\div}}
\newcommand{\isasymcdot}{\isamath{\cdot}}
\newcommand{\isasymstar}{\isamath{\star}}
\newcommand{\isasymbullet}{\boldmath\isamath{\mathchoice{\displaystyle{\cdot}}{\textstyle{\cdot}}{\scriptstyle{\bullet}}{\scriptscriptstyle{\bullet}}}}
\newcommand{\isasymcirc}{\isamath{\circ}}
\newcommand{\isasymdagger}{\isamath{\dagger}}
\newcommand{\isasymddagger}{\isamath{\ddagger}}
\newcommand{\isasymlhd}{\isamath{\lhd}} %requires amssymb
\newcommand{\isasymrhd}{\isamath{\rhd}} %requires amssymb
\newcommand{\isasymunlhd}{\isamath{\unlhd}} %requires amssymb
\newcommand{\isasymunrhd}{\isamath{\unrhd}} %requires amssymb
\newcommand{\isasymtriangleleft}{\isamath{\triangleleft}}
\newcommand{\isasymtriangleright}{\isamath{\triangleright}}
\newcommand{\isasymtriangle}{\isamath{\triangle}}
\newcommand{\isasymtriangleq}{\isamath{\triangleq}} %requires amssymb
\newcommand{\isasymoplus}{\isamath{\oplus}}
\newcommand{\isasymOplus}{\isamath{\bigoplus\,}}
\newcommand{\isasymotimes}{\isamath{\otimes}}
\newcommand{\isasymOtimes}{\isamath{\bigotimes\,}}
\newcommand{\isasymodot}{\isamath{\odot}}
\newcommand{\isasymOdot}{\isamath{\bigodot\,}}
\newcommand{\isasymominus}{\isamath{\ominus}}
\newcommand{\isasymoslash}{\isamath{\oslash}}
\newcommand{\isasymdots}{\isamath{\dots}}
\newcommand{\isasymcdots}{\isamath{\cdots}}
\newcommand{\isasymSum}{\isamath{\sum\,}}
\newcommand{\isasymProd}{\isamath{\prod\,}}
\newcommand{\isasymCoprod}{\isamath{\coprod\,}}
\newcommand{\isasyminfinity}{\isamath{\infty}}
\newcommand{\isasymintegral}{\isamath{\int\,}}
\newcommand{\isasymointegral}{\isamath{\oint\,}}
\newcommand{\isasymclubsuit}{\isamath{\clubsuit}}
\newcommand{\isasymdiamondsuit}{\isamath{\diamondsuit}}
\newcommand{\isasymheartsuit}{\isamath{\heartsuit}}
\newcommand{\isasymspadesuit}{\isamath{\spadesuit}}
\newcommand{\isasymaleph}{\isamath{\aleph}}
\newcommand{\isasymemptyset}{\isamath{\emptyset}}
\newcommand{\isasymnabla}{\isamath{\nabla}}
\newcommand{\isasympartial}{\isamath{\partial}}
\newcommand{\isasymRe}{\isamath{\Re}}
\newcommand{\isasymIm}{\isamath{\Im}}
\newcommand{\isasymflat}{\isamath{\flat}}
\newcommand{\isasymnatural}{\isamath{\natural}}
\newcommand{\isasymsharp}{\isamath{\sharp}}
\newcommand{\isasymangle}{\isamath{\angle}}
\newcommand{\isasymcopyright}{\isatext{\rm\copyright}}
\newcommand{\isasymregistered}{\isatext{\rm\textregistered}}
\newcommand{\isasymhyphen}{\isatext{\rm-}}
\newcommand{\isasyminverse}{\isamath{{}^{-1}}}
\newcommand{\isasymonesuperior}{\isamath{\mathonesuperior}} %requires latin1
\newcommand{\isasymonequarter}{\isatext{\rm\textonequarter}} %requires latin1
\newcommand{\isasymtwosuperior}{\isamath{\mathtwosuperior}} %requires latin1
\newcommand{\isasymonehalf}{\isatext{\rm\textonehalf}} %requires latin1
\newcommand{\isasymthreesuperior}{\isamath{\maththreesuperior}} %requires latin1
\newcommand{\isasymthreequarters}{\isatext{\rm\textthreequarters}} %requires latin1
\newcommand{\isasymordfeminine}{\isatext{\rm\textordfeminine}}
\newcommand{\isasymordmasculine}{\isatext{\rm\textordmasculine}}
\newcommand{\isasymsection}{\isatext{\rm\S}}
\newcommand{\isasymparagraph}{\isatext{\rm\P}}
\newcommand{\isasymexclamdown}{\isatext{\rm\textexclamdown}}
\newcommand{\isasymquestiondown}{\isatext{\rm\textquestiondown}}
\newcommand{\isasymeuro}{\isatext{\textgreek{\euro}}} %requires greek babel
\newcommand{\isasympounds}{\isamath{\pounds}}
\newcommand{\isasymyen}{\isatext{\yen}} %requires amssymb
\newcommand{\isasymcent}{\isatext{\textcent}} %requires textcomp
\newcommand{\isasymcurrency}{\isatext{\textcurrency}} %requires textcomp
\newcommand{\isasymdegree}{\isatext{\rm\textdegree}} %requires latin1
\newcommand{\isasymamalg}{\isamath{\amalg}}
\newcommand{\isasymmho}{\isamath{\mho}} %requires amssymb
\newcommand{\isasymlozenge}{\isamath{\lozenge}} %requires amssymb
\newcommand{\isasymwp}{\isamath{\wp}}
\newcommand{\isasymwrong}{\isamath{\wr}}
\newcommand{\isasymstruct}{\isamath{\diamond}}
\newcommand{\isasymacute}{\isatext{\'\relax}}
\newcommand{\isasymindex}{\isatext{\i}}
\newcommand{\isasymdieresis}{\isatext{\"\relax}}
\newcommand{\isasymcedilla}{\isatext{\c\relax}}
\newcommand{\isasymhungarumlaut}{\isatext{\H\relax}}
\newcommand{\isasymspacespace}{\isamath{~~}}
\newcommand{\isasymmodule}{\isamath{\langle}\isakeyword{module}\isamath{\rangle}}
\newcommand{\isasymsome}{\isamath{\epsilon\,}}