\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*{nonstandard analysis}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{an_approach_via_ultrafilters}{An approach via ultrafilters}\dotfill \pageref*{an_approach_via_ultrafilters} \linebreak
\noindent\hyperlink{filterquotients_of_topoi}{Filterquotients of topoi}\dotfill \pageref*{filterquotients_of_topoi} \linebreak
\noindent\hyperlink{sheaves_on_the_topos_of_filters}{Sheaves on the topos of filters}\dotfill \pageref*{sheaves_on_the_topos_of_filters} \linebreak
\noindent\hyperlink{measure_and_generalized_functions}{Measure and generalized functions}\dotfill \pageref*{measure_and_generalized_functions} \linebreak
\noindent\hyperlink{ToposModels}{Relationship to other types of infinitesimal}\dotfill \pageref*{ToposModels} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

Nonstandard analysis is a rich formalization of [[analysis]] that uses a certain explicit notions of [[infinitesimal objects]]. In fact, not only infinitesimal but also infinitely large can be accomodated (and must be). Moreover, not only the field of [[real numbers]], but more general algebraic structures can be extended, essentially via a construction of [[ultraproducts]]; also general sets can be extended to contain nonstandard elements (see [[internal set]]).

In fact the nonstandard method is not limited to analysis, but is rather a method of producing a models in the sense of [[model theory]], as pioneered by Robinson or using a syntactic extension of set theory, like the theory of [[internal sets]] by Nelson.

See also [[nonstandard analysis in topology]], [[internal set]].

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

At its beginning, infinitesimal calculus was developed nonrigorously, though many interesting arguments and formal manipulations were found. Cauchy and Weierstra\ss{} introduced the $\epsilon$-$\delta$ approach, which enabled modern rigorous [[analysis]], but sometimes this method is cumbersome. For example, sometimes one needs to work with several infinitesimal levels or kinds of continuity in the same problem, and finding estimates may be very cumbersome. One would like to introduce infinitesimal quantities as additional elements of the sets of usual (`standard') quantities. Several related rigorous frameworks appeared under the name of \textbf{nonstandard analysis}, since the first such discovered by [[Abraham Robinson]]. Most often, approaches using [[ultrafilters]], certain classes called [[internal sets]] and using [[topos theory]] enable the foundation of nonstandard analysis. Many properties and theorems from classical analysis imply new statements of nonstandard analysis; the mechanism is the so-called \textbf{transfer principle}, which can be required axiomatically without respect to a particular model of nonstandard analysis.

\hypertarget{models}{}\subsection*{{Models}}\label{models}

\hypertarget{an_approach_via_ultrafilters}{}\subsubsection*{{An approach via ultrafilters}}\label{an_approach_via_ultrafilters}

Assuming the [[axiom of choice]] (whose full strength is not necessary), there exists a free (= not containing finite subsets) [[ultrafilter]] $F$ on the set of [[natural numbers]] $\mathbb{N}=\{1,2,3,\ldots\}$, and such ultrafilters are in $1$--$1$ correspondence with finitely additive [[measures]] on $\mathbb{N}$ (using the [[power set|algebra of all subsets]]) taking values in the two element set $\{0,1\}$.

Fix a free ultrafilter $F$ on $\mathbb{N}$, and consider the set of all [[sequences]] of real numbers, $\mathbb{R}^{\mathbb{N}}$. So an element in here is a sequence

\begin{displaymath}
a = (a_0, a_1, a_2, \cdots)
\end{displaymath}
of real numbers.

We write $f\sim_F g$ if the set $\{i\in\mathbb{N}|f(i)=g(i)\}$ belongs to $F$; these are precisely the sequences which are equal almost everywhere with respect to the associated measure. The relation $\sim_F$ is an [[equivalence relation]] and ${}^*\mathbb{R} := \mathbb{R}^{\mathbb{N}}/{\sim_F}$ is a \textbf{nonstandard extension} of $\mathbb{R}$, whose elements are sometimes called \textbf{hyperreal numbers}.

(This is a special case of the [[ultraproduct]] construction in [[model theory]]. In fact, we could have started with an ultrafilter on any set, not just $\mathbb{N}$. Such more general ultraproducts are necessary in order to obtain more refined models of nonstandard analysis satisfying stronger ``saturation'' principles.)

Given $f\in \mathbb{R}^{\mathbb{N}}$, we write $f_F$ for its equivalence class in ${}^*\mathbb{R}$. In particular, given any real number $r\in \mathbb{R}$ the image $^* r :=(i\mapsto r)_F$ of the constant sequence

\begin{displaymath}
r = (r,r,r, \cdots )
  \,.
\end{displaymath}
is an element in $^*\mathbb{R}$ and this gives an [[injection]] $*:\mathbb{R}\hookrightarrow {}^*\mathbb{R}$.

$^*\mathbb{R}$ is equipped with a linear ordering given by

\begin{displaymath}
f_F\lt g_F \;\Leftrightarrow\; \{i\in\mathbb{N}|f(i)\lt g(i)\}\in F,
\end{displaymath}
which makes $*:\mathbb{R}\hookrightarrow {}^*\mathbb{R}$ a [[monotone function]].

\begin{itemize}%
\item An element $\delta\in{}^*\mathbb{R}$ such that $^* 0\lt\delta$ is called a \textbf{positive number} .


\item An element $\delta$ such that for all positive $r\in \mathbb{R}$ we have $-{}^* r \lt  \delta\lt{}^* r$ is called an \textbf{infinitesimal number}.



\end{itemize}
Unlike in the real numbers, positive infinitesimal numbers exist: for example the class $f_F$ where $f:n\mapsto 1/n$ is such and $g_F$ for $g:n\mapsto 1/n^2$ is a different one.

Let $n$ be a nonnegative integer and $u:\mathbb{R}^n\to\mathbb{R}$ a function. Then there is a nonstandard extension $^* u:({}^*\mathbb{R})^n\to{}^*\mathbb{R}$ of $u$; it is defined by

\begin{displaymath}
^*u(f^1_F,\ldots, f^n_F) = g \;\Leftrightarrow\; \{i\in\mathbb{N}| u(f^1(i),\ldots,f^n(i))=g(i) \}\in F.
\end{displaymath}
This is indeed an extension of $u$ in the sense that $^* u({}^*r_1,\ldots,{}^*r_n)={}^* r$ iff $u(r_1,\ldots,r_n)=r$. This way, the usual operations $+,\cdot$ and the absolute value $|\cdot|$ extend to $^*\mathbb{R}$; usually one denotes these and other standard operations on ${}*\mathbb{R}$ without putting $^*$ in front, writing simply e.g. $f_F+g_F$.

To extend division appropriately, we need a little bit more care as it is originally just partially defined, so we need an extension of the formalism to subsets of the real line. In particular there is a following definition of an extension $^* E\subset{}^*\mathbb{R}$ of a subset $E\subset\mathbb{R}$:

\begin{displaymath}
f_F\in{}^* E \;\Leftrightarrow\; \{i\in\mathbb{N}|f(i)\in E\}\in F.
\end{displaymath}
(For example, a number is positive, as defined earlier, if an only if it belongs to $^*\{r|r \gt 0\}$.) Then division is extended to a function $^*\mathbb{R} \times ^*\{r|r \ne 0\} \to ^*\mathbb{R}$. If $1/x$ is infinitesimal, then $x$ itself is \textbf{infinite}.

Conversely, an element $x\in{}^*\mathbb{R}$ is \textbf{finite} if $|x|\lt {}^* r$ for some $r\in\mathbb{R}$. Every finite element $x\in{}^*\mathbb{R}$ is infinitely close to a unique real number $q\in\mathbb{R}$ in the sense that $x-{}^*q$ is infinitesimal. We say that $q$ is the \textbf{standard part} of $x$ and is denoted by $q= st(x)$. Given a real number $r\in\mathbb{R}$, the subset $\mu(r)$ of all elements $x\in{}^*\mathbb{R}$ such that $st(x)=r$ is said to be the \textbf{monad} of the real number $r\in\mathbb{R}$. Monads should be thought of as infinitesimal neighborhoods. An elementary fact: a subset $E\subset\mathbb{R}$ is open iff $\mu(r)\subset{}^*E$ for all $r\in E$; $E$ is closed iff $st(x)\in E$ for all finite $x\in{}^* E$; and $E$ is [[compact space|compact]] iff, for all $x\in{}^* E$, $x$ is finite and $st(x)\in E$.

In this model of nonstandard analysis, the [[transfer principle]] is a corollary of a general theorem on ultraproducts due Jerzy o. It can be stated in terms of a certain formal language $L(\mathbb{R})$ of the real numbers. We can also extend this model to ultrapowers of larger sets, not just $\mathbb{R}$ itself, with a corresponding extension of the language. In the limit where we reach an entire ``universe'' of mathematics, this leads to the topos-theoretic filterquotient and sheaf models below.

\hypertarget{filterquotients_of_topoi}{}\subsubsection*{{Filterquotients of topoi}}\label{filterquotients_of_topoi}

The ultrapower construction above can be performed in the general context of [[topos]] theory. From any topos $\mathcal{E}$ and any filter $\Phi$ of [[subterminal objects]] in $\mathcal{E}$, one can construct a topos $\mathcal{E}/\Phi$, the [[filterquotient]] of $\mathcal{E}$ by $\Phi$. There is a [[logical functor]] $\mathcal{E} \to \mathcal{E}/\Phi$.

If $\mathcal{E} = Set / \mathbb{N}$, then any filter on $\mathbb{N}$ gives a filter of subterminals in $\mathcal{E}$, whose corresponding filterquotient corresponds to the above construction. The composite functor

\begin{displaymath}
Set \to Set/\mathbb{N} \to (Set/\mathbb{N})/\Phi
\end{displaymath}
might be written $^*(-)$. If $\Phi$ is an ultrafilter, then $(Set/\mathbb{N})/\Phi$ is a two-valued topos, whose [[internal logic]] is essentially that of the ultrafilter model described above. In particular, the global elements of $^*\mathbb{R}$, as an object of this topos, are precisely the ``hyperreal numbers'' described above.

In this context, the [[transfer principle]] is the fact that the functor $^*(-)$ is both [[logical functor|logical]] and [[conservative functor|conservative]], and hence it both preserves and reflects the truth of formulas in the internal languages.

\hypertarget{sheaves_on_the_topos_of_filters}{}\subsubsection*{{Sheaves on the topos of filters}}\label{sheaves_on_the_topos_of_filters}

A different topos-theoretic construction is to consider the topos of sheaves on a category of filters. This topos models the [[internal set]] theory of Nelson, a more axiomatic approach to nonstandard analysis. References:

\begin{itemize}%
\item I. Moerdijk, \emph{A model for intuitionistic non-standard arithmetic}
\item Erik Palmgren, \emph{\href{http://www.sciencedirect.com/science/article/pii/S0168007296000413}{A sheaf-theoretic foundation for nonstandard analysis}}
\item E. Palmgren, \emph{A constructive approach to nonstandard analysis}
\item E. Palmgern, \emph{Remarks on a constructive sheaf model of nonstandard analysis}, \href{http://dx.doi.org/10.1.1.30.4785}{doi}
\item Jonas Eliasson, \emph{Ultrapowers as sheaves on a category of ultrafilters}

\end{itemize}
\hypertarget{measure_and_generalized_functions}{}\subsection*{{Measure and generalized functions}}\label{measure_and_generalized_functions}

The Lebesgue measure on $\mathbf{R}^n$ extends to [[Loeb measure]] on ${}^\ast\mathbf{R}^n$. This may be used for probability theory and also for generalized functions.

The theory of generalized functions of Schwarz can be reproduced by nonstandard analysis:

\textbf{Theorem.} ([[Abraham Robinson]]) Every generalized function $f:\mathbf{R}\to\mathbf{R}$ can be represented as the integration of the product of test function with a nonstandard smooth function $\tilde{f}: {}^\ast\mathbf{R}\to{}^\ast \mathbf{R}$

\begin{displaymath}
(f,\phi) = \int \tilde{f} \phi d x
\end{displaymath}
There is also inutionistic version of nonstandard analysis approach to generalized functions as well as nonstandard approaches to Sato [[hyperfunction]]s (Sousa pinto), to Coulombeu distributions etc.

\hypertarget{ToposModels}{}\subsection*{{Relationship to other types of infinitesimal}}\label{ToposModels}

There are other ways of realizing the notion of [[infinitesimal number]] precisely, such as [[synthetic differential geometry]] and the [[surreal number]]s. Neither seem to be very closely related to NSA---the techniques and flavor of each subject are quite different. However, some things can be said.

\begin{itemize}%
\item While the most common infinitesimals appearing in SDG are [[nilpotent]], in contrast to those of NSA which are invertible, some models of SDG do contain invertible infinitesimals; see \href{http://ncatlab.org/nlab/show/infinitesimal+number#NSAvsSDG}{here}.


\item Since the surreal numbers are the universally embedding ordered field, any field of hyperreals can be embedded in the surreals. However, such embeddings don't seem very useful, since they don't preserve any of the important structure of the hyperreals (such as the transfer principle).



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\begin{itemize}%
\item [[Sergio Albeverio]], Jens Erik Fenstad, Raphael Hoegh-Krohn, \emph{Nonstandard methods in stochastic analysis and mathematical physics}, Academic Press 1986 (there is also a Dover 2009 edition and a 1990 Russian translation)


\item [[Jerome Keisler]], \emph{Elementary calculus: an infinitesimal approach}, \href{http://www.math.wisc.edu/~keisler/calc.html}{online} undergraduate textbook.


\item [[Abraham Robinson]], \emph{Non-standard analysis}, 1966


\item Wikipedia: \href{http://en.wikipedia.org/wiki/Non-standard_analysis}{nonstandard analysis}, \href{http://en.wikipedia.org/wiki/Ultraproduct}{ultraproduct}, \href{http://en.wikipedia.org/wiki/Hyperreal_numbers}{hyperreal numbers}, \href{http://en.wikipedia.org/wiki/Abraham_Robinson}{Abraham Robinson}, \href{http://en.wikipedia.org/wiki/Constructive_non-standard_analysis}{constructive non-standard analysis}, \href{http://en.wikipedia.org/wiki/Criticism_of_non-standard_analysis}{criticism of nonstandard analysis}


\item Sergio Salbany, Todor Todorov, \emph{Nonstandard analysis in topology}, \href{http://arxiv.org/abs/1107.3323}{arxiv/1107.3323}


\item [[Ieke Moerdijk]], \emph{A model for intuitionistic nonstandard arithmetic}, Annals of Pure and Applied Logic \textbf{73} (1995), pp. 37--51.


\item [[Jaap van Oosten]], \emph{Synthetic Nonstandard Arithmetic}, 2011 (\href{https://www.staff.science.uu.nl/~ooste110/talks/darmstadt2.pdf}{pdf})


\item Juha Ruokolainen, \emph{Constructive nonstandard analysis without actual infinity}, 2004, \href{https://oa.doria.fi/bitstream/handle/10024/2865/construc.pdf}{pdf}


\item E. Palmgren, \emph{Developments in Constructive Nonstandard Analysis}, Bull. Symbolic Logic \textbf{4}, n. 3 (1998), 233--272.


\item E. Palmgren, \emph{Constructive nonstandard representations of generalized functions}, 


\item Robert A. Herrmann, \emph{Nonstandard analysis and generalized functions}, \href{http://arxiv.org/abs/math/0403303}{math.FA/0403303}


\item Robert A. Hermann, \emph{Nonstandard analysis applied to advanced undergraduate mathematics}, \href{http://www.arXiv.org/abs/math.GM/0312432}{math.GM/0312432}


\item A. E. Hurd, P. A. Loeb, \emph{Introduction to nonstandard real analysis}, Acad. Press 1985.


\item Hans Vernaeve, \emph{Nonstandard principles for generalized functions}, \href{http://arxiv.org/abs/1101.6075}{arxiv/1101.6075}


\item Imme van den Bergh, V\'i{}tor Manuel Carvalho das Neves (eds.), \emph{The strength of nonstandard analysis}, \href{http://books.google.com/books?id=nWyPEAW16U8C}{gBooks}


\item R. F. Hoskins, J. Sousa Pinto, \emph{Theories of generalized functions}, Horwood Publ. 2005


\item Imme van den Berg, \emph{Nonstandard asymptotic analysis}, Lec. Notes Math. \textbf{1249}, Springer 1987


\item Edward Nelson, \emph{Radically elementary probability theory}


\item Diener-Diener (eds.), \emph{Nonstandard analysis in practice}


\item Leob-Wolff (eds.), \emph{Nonstandard analysis for the working mathematician}


\item [[Terry Tao]], \emph{Ultraproducts as a bridge between discrete and continuous analysis} (\href{http://terrytao.wordpress.com/2013/12/07/ultraproducts-as-a-bridge-between-discrete-and-continuous-analysis/}{web})


\item V. A. Lyubetski, \emph{  .     }, Uspekhi Mat. Nauk \textbf{44} (1989), no. 4(268), 99--153, 256; translation \emph{Valuations and sheaves. On some questions of non-standard analysis}, in Russian Math. Surveys \textbf{44} (1989), no. 4, 37--112 \href{http://www.ams.org/mathscinet-getitem?mr=1023104}{MR1023104} \href{http://dx.doi.org/10.1070/RM1989v044n04ABEH002140}{doi} \href{http://iopscience.iop.org/0036-0279/44/4/R03/pdf/0036-0279_44_4_R03.pdf}{IOP pdf} \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=1849&what=fullt&option_lang=rus}{rus pdf}


\item Bruno Dinis, \emph{Nonstandard intuitionistic interpretations}, \href{http://arxiv.org/abs/1512.07113}{arxiv/1512.07113}


\item Sam Sanders, \emph{The unreasonable effectiveness of nonstandard analysis}, \href{http://arxiv.org/abs/1508.07434}{arxiv/1508.07434}


\item V. Kanovei, \emph{A course on foundations of nonstandard analysis}, (With a preface by M. Reeken.), IPM, Tehran, Iran, 1994.


\item V. Kanovei, M. Reeken, Internal approach to external sets and uni- verses, Part 1, Bounded set theory, Studia Logica, 1995.



\end{itemize}
The relation of the techniques of the pioneers of infinitesimal calculus and the modern nonstandard analysis is discussed in

\begin{itemize}%
\item Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz, Mikhail G. Katz, Semen S. Kutateladze, David Sherry, \emph{Toward a history of mathematics focused on procedures}, \href{http://arxiv.org/abs/1609.04531}{arxiv/1609.04531}

\end{itemize}
[[!redirects hyperreal]] [[!redirects hyperreal number]] [[!redirects hyperreals]] [[!redirects hyperreal numbers]] [[!redirects NSA]]



\end{document}