\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{null subset}

\hypertarget{null_and_full_sets}{}\section*{{Null and full sets}}\label{null_and_full_sets}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{in_a_measure_space}{In a measure space}\dotfill \pageref*{in_a_measure_space} \linebreak
\noindent\hyperlink{in_untraditional_measurable_spaces}{In untraditional measurable spaces}\dotfill \pageref*{in_untraditional_measurable_spaces} \linebreak
\noindent\hyperlink{in_smooth_manifolds}{In smooth manifolds}\dotfill \pageref*{in_smooth_manifolds} \linebreak
\noindent\hyperlink{logic_of_fullnull_sets}{Logic of full/null sets}\dotfill \pageref*{logic_of_fullnull_sets} \linebreak


([[null set|Null set]] redirects here; for the notion in set theory, see [[empty set]].)

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[measure theory]], a \emph{null set} is a [[subset]] of a [[measure space]] (or [[measurable space]]) that is so small that it can be neglected: it might as well be the [[empty subset]]; its measure is [[zero]]. Similarly, a \emph{full set} is a subset that is so large that it might as well be the [[improper subset]] (the entire space). One also says that a null set has \emph{null measure} and a full set has \emph{full measure}.

Traditionally, full sets are not usually referred to explicitly; in [[classical mathematics]], they are simply the [[complements]] of null sets. However, they are often referred to implicitly through such terminology as `almost everywhere'. Also, in [[constructive mathematics]], full sets are more fundamental than null sets; they are not simply the complements of the latter.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

The definitions depend on the context.

\hypertarget{in_a_measure_space}{}\subsubsection*{{In a measure space}}\label{in_a_measure_space}

In a traditional [[measure space]], we have an abstract [[set]] $X$, a $\sigma$-[[sigma-algebra|algebra]] (or similar structure) $\mathcal{M}$ consisting of the [[measurable subsets]] of $X$, and a [[measure]] $\mu$ mapping each measurable set $A$ to a [[real number]] (or similar quantity) $\mu(A)$, the measure of $A$.

A measurable subset $B$ of $X$ is \textbf{full} if, given any measurable set $A$, $\mu(A \cap B) = \mu(A)$; an arbitrary subset of $X$ is \textbf{full} if it's a [[superset]] of a full measurable set. [[de Morgan duality|Dually]], a measurable set $B$ is \textbf{null} if, given any measurable set $A$, $\mu(A \cup B) = \mu(A)$; an arbitrary subset of $X$ is \textbf{null} if it's a [[subset]] of a null measurable set.

Some equivalent characterisations ([[constructive mathematics|constructively]] valid for measures on [[Cheng spaces]] except as stated):

\begin{itemize}%
\item A measurable set $B$ is null iff $\mu(C) = 0$ for every measurable subset of $B$.
\item If $\mu$ is a [[positive measure]], then a measurable set $B$ is null iff $\mu(B) = 0$.
\item If $\mu$ is a [[finite measure]] with total measure $I$, then a measurable set $B$ is full iff $\mu(C) = I$ for every measurable superset of $B$.
\item If $\mu$ is both positive and finite (so a [[probability measure]] up to rescaling), then a measurable set $B$ is full iff $\mu(B) = I$.
\item If $\mu$ is [[complete measure|complete]], then every null set is measurable and every full set is measurable (which is basically the definition of `complete') and consequently the preceding properties continue to hold when the adjective `measurable' is removed.
\item Using [[excluded middle]], a set is null iff its complement is full, and a set is full iff its complement is null. (Even constructively, if a set is null, then its complement is full.)
\item Even constructively, a measurable set is null iff its measurable complement (the complement in the algebraic structure of complemented pairs in a Cheng measurable space) is full, and a measurable set is full iff its measurable complement is null.

\end{itemize}
\hypertarget{in_untraditional_measurable_spaces}{}\subsubsection*{{In untraditional measurable spaces}}\label{in_untraditional_measurable_spaces}

Traditionally, a [[measurable space]] is simply an abstract [[set]] $X$ and a $\sigma$-[[sigma-algebra|algebra]] (or similar structure) $\mathcal{M}$ consisting of the [[measurable subsets]] of $X$. There is no notion of null or full subsets of such a space. However, there are two (essentially equivalent) variations of this concept in which null and full subsets do make sense.

One variation is to simply equip the space with a $\delta$-[[delta-filter|filter]] of measurable subsets, which are declared to be the full measurable subsets. Then a general \textbf{full subset} is a superset of a measurable full subset, and a \textbf{null subset} is any set whose complement is full. (Alternatively, equip the space with a $\sigma$-[[sigma-ideal|ideal]] of measurable subsets, which are declared to be the null measurable subsets.) In particular, a [[localizable measurable space]] is a measurable space so equipped such that the [[Boolean algebra]] of measurable sets modulo null sets (or modulo full sets if this is done by identifying the full sets with $X$) is [[complete boolean algebra|complete]].

Another variation, used especially in [[constructive mathematics]], is a [[Cheng measurable space]]. This consists of a set $X$ equipped with a $\sigma$-semialgebra of [[disjoint subsets|disjoint]] \emph{pairs} of subsets of $X$, declared to be the \emph{complemented pairs}. A set is measurable iff it appears as one component of a complemented pair. A measurable subset is \emph{full} if it appears as one component of a complemented pair whose other component is [[empty subset|empty]], or equivalently (given the structure of the algebra of complemented pairs) if it is the [[union]] of the two components of any complemented pair. Then a general \textbf{full subset} is a superset of a measurable full subset, and a \textbf{null subset} is any set whose complement is full.

These are actually equivalent concepts. Given a measurable space equipped with a $\delta$-filter of measurable full subsets, define a complemented pair to be a pair of disjoint measurable subsets whose union is full. Conversely, given a Cheng measurable space, the measurable subsets and measurable full subsets as defined above comprise a $\sigma$-algebra and a $\delta$-filter in it. (But constructively, the algebra of measurable subsets, while closed under the appropriate operations, will generally not be a [[boolean algebra]].)

\hypertarget{in_smooth_manifolds}{}\subsubsection*{{In smooth manifolds}}\label{in_smooth_manifolds}

A subset $A$ of an $n$-dimensional [[smooth manifold]] $X$ is \textbf{null} or \textbf{full} (respectively) if its [[preimage]] under every [[coordinate chart]] is a null or full subset (respectively) of the chart's domain (which is an [[open subset]] of the [[Cartesian space]] $\mathbb{R}^n$) under [[Lebesgue measure]].

This is actually better behaved than it may at first seem. If $A$ is [[open cover|covered]] by an [[atlas]] $(\phi_i\colon U_i \to X)_i$, then $A$ is null or full as soon as $\phi_i^*(A)$ is null/full in $U_i$ for every index $i$. In particular, if $A$ is contained in a single coordinate chart (which is not very likely for a full set but fairly common for null sets), then it is sufficient to check its preimage under that one. This fact depends on the smoothness and fails for [[topological manifolds]].

As we can define a [[measurable subset]] of a smooth manifold similarly, this means that every smooth manifold gives rise to a measurable space equipped with a $\delta$-filter of full subsets (and hence to a Cheng measurable space); this space is always localizable.

(Details? Is $C^1$ sufficient? Conversely, is paracompactness necessary to keep the covers manageable?)

\hypertarget{logic_of_fullnull_sets}{}\subsection*{{Logic of full/null sets}}\label{logic_of_fullnull_sets}

A property of elements of $X$ (given by a [[subset]] $S$ of $X$) can be considered modulo null sets. We say that the property $\phi$ is true \textbf{almost everywhere} or \textbf{almost always} if it is true on some full set, that is if $\{X | \phi\}$ is full. Dually, we say that $\phi$ is true \textbf{almost nowhere} or \textbf{almost never} if $\{X | \phi\}$ is null. It is better to use the [[negation]] of `almost nowhere', although the terminology for this is not really standard; say that $\phi$ is true \textbf{somewhere significant} if $\{X | \phi\}$ is non-null.

Note that being true almost everywhere is a weakening of being true everywhere (given by the [[universal quantifier]] $\forall$), while being true somewhere significant is a strengthening of being true somewhere (given by the [[particular quantifier]] $\exists$). Indeed we can build a logic out of these. Use $\ess\forall i, \phi[i]$ or $\ess\forall \phi$ to mean that a [[predicate]] $\phi$ on $X$ is true almost everywhere, and use $\ess\exists i, \phi[i]$ or $\ess\exists \phi$ to mean that $\phi$ is true somewhere significant. Then we have:

\begin{displaymath}
\forall \phi \;\Rightarrow\; \ess\forall \phi
\end{displaymath}
\begin{displaymath}
\ess\exists \phi \;\Rightarrow\; \exists \phi
\end{displaymath}
\begin{displaymath}
\ess\forall (\phi \wedge \psi) \;\Leftrightarrow\; \ess\forall \phi \wedge \ess\forall \psi
\end{displaymath}
\begin{displaymath}
\ess\exists (\phi \wedge \psi) \;\Rightarrow\; \ess\exists \phi \wedge \ess\exists \psi
\end{displaymath}
\begin{displaymath}
\ess\forall (\phi \vee \psi) \;\Leftarrow\; \ess\forall \phi \wedge \ess\forall \psi
\end{displaymath}
\begin{displaymath}
\ess\exists (\phi \vee \psi) \;\Leftrightarrow\; \ess\exists \phi \vee \ess\exists \psi
\end{displaymath}
\begin{displaymath}
\ess\forall \neg{\phi} \;\Leftrightarrow\; \neg\ess\exists \phi
\end{displaymath}
and other analogues of theorems from [[predicate logic]]. Note that the last item listed requires [[excluded middle]] even though its analogue from ordinary predicate logic does not.

A similar logic is satisfied by `[[eventually]]' and its dual (`frequently') in [[filters]] and [[nets]].

[[!redirects null set]] [[!redirects null sets]] [[!redirects null subset]] [[!redirects null subsets]] [[!redirects null measure]] [[!redirects set of null measure]] [[!redirects sets of null measure]] [[!redirects subset of null measure]] [[!redirects subsets of null measure]]

[[!redirects full set]] [[!redirects full sets]] [[!redirects full subset]] [[!redirects full subsets]] [[!redirects full measure]] [[!redirects set of full measure]] [[!redirects sets of full measure]] [[!redirects subset of full measure]] [[!redirects subsets of full measure]]

[[!redirects almost everywhere]] [[!redirects almost nowhere]] [[!redirects somewhere significant]]



\end{document}