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\begin{document}

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\section*{measure theory}

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

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{some_subfields_and_applications}{Some subfields and applications}\dotfill \pageref*{some_subfields_and_applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Measure theory studies [[measurable spaces]] and [[measure spaces]].

\hypertarget{history}{}\subsection*{{History}}\label{history}

Measure theory is the field of mathematics that grew out of the [[Lebesgue measure|Lebesgue integral]] and Kolmogorov's axioms for [[probability]].

\hypertarget{some_subfields_and_applications}{}\subsection*{{Some subfields and applications}}\label{some_subfields_and_applications}

The general measure theory studies general notions and constructions in measure theory, like the connection to integration, the measure spaces, derivation by measure, [[Caratheodory construction]] and so on.

[[probability theory|Probability theory]] studies special class of measures, so called [[probability measure]]s which are normalized to unity.

Measure theory is very much having a central role in studying so called [[ergodic theory]] of dynamical system.

[[geometric measure theory|Geometric measure theory]] is the geometric study of measures of subsets of Euclidean space and the measure theoretic aspects of various geometric objects, like the integration of classes of currents and their extremization properties.

There is a generalization, the noncommutative measure theory, which is more or less the study of [[von Neumann algebra]], see \hyperlink{Connes1995}{Connes (1995)}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[expectation value]]


\item [[valuation (measure theory)]]



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

\begin{itemize}%
\item Ernst-Erich Doberkat, \emph{Measures and all that --- A Tutorial}, \href{http://arxiv.org/abs/1409.2662}{arxiv/1409.2662}


\item [[Alain Connes]] (1995); \emph{[[Noncommutative Geometry]]}.


\item D.H. Fremlin (2001); \emph{Measure Theory}; 5 volumes, \href{http://www.essex.ac.uk/maths/people/fremlin/mt.htm}{web}.



\end{itemize}
Discussion via [[Boolean toposes]] is in

\begin{itemize}%
\item [[Simon Henry]], \emph{Measure theory over boolean toposes}, Mathematical Proceedings of the Cambirdge Philosophical Society, 2016 (\href{https://arxiv.org/abs/1411.1605}{arXiv:1411.1605})

\end{itemize}


\end{document}