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\begin{document}

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\section*{Radon–Nikodym derivative}

\hypertarget{radonnikodym_derivatives}{}\section*{{Radon--Nikodym derivatives}}\label{radonnikodym_derivatives}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{notation}{Notation}\dotfill \pageref*{notation} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given two [[measures]] $\mu, \nu$ on the same [[measurable space]], their Radon--Nikodym derivative is essentially their ratio $\mu/\nu$, although this is traditionally written $\mathrm{d}\mu/\mathrm{d}\nu$ because of analogies with [[differentiation]]. This ratio or derivative is a [[measurable function]] which is defined up to [[almost equality|equality almost everywhere]] with respect to the divisor $\nu$. It only exists iff $\mu$ is [[absolutely continuous measure|absolutely continuous]] with respect to $\nu$.

Integration on a general [[measure space]] can be seen as the process of multiplying a measure by a function to get a measure. Then the Radon--Nikodym derivative is the reverse of this: dividing two measures to get a function.

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

Let $X$ be a [[measurable space]] (so $X$ consists of a [[set]] ${|X|}$ and a $\sigma$-[[sigma-algebra|algebra]] $\mathcal{M}_X$), and let $\mu$ and $\nu$ be [[measures]] on $X$, valued in the [[real numbers]] (and possibly taking infinite values) or in the [[complex numbers]] (and taking only finite values). Let $f$ be a [[measurable function]] $f$ (with real or complex values) on $X$.

\begin{udefn}
The function $f$ is a \textbf{Radon--Nikodym derivative} of $\mu$ with respect to $\nu$ if, given any [[measurable subset]] $A$ of $X$, the $\mu$-measure of $A$ equals the integral of $f$ on $A$ with respect to $\nu$:

\begin{displaymath}
\mu(A) = \int_A f \nu = \int_{x \in A} f(x) \mathrm{d}\nu(x) .
\end{displaymath}
(The latter two expressions in this equation are different notations for the same thing.)

\end{udefn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

These properties are basic to the concept; the notation is as in the definition above.

\begin{theorem}
\label{}\hypertarget{}{}
Let $f$ be a Radon--Nikodym derivative of $\mu$ with respect to $\nu$, and let $g$ be a measurable function on $X$. Then $g$ is a Radon--Nikodym derivative of $\mu$ with respect to $\nu$ if and only if $f$ and $g$ are equal almost everywhere with respect to $\nu$.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
If a Radon--Nikodym derivative of $\mu$ with respect to $\nu$ exists, then $\mu$ is [[absolutely continuous measure|absolutely continuous]] with respect to $\nu$.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
If $\mu$ is [[absolutely continuous measure|absolutely continuous]] with respect to $\nu$ and both $\mu$ and $\nu$ are $\sigma$-[[sigma-finite measure|finite]], then a Radon--Nikodym derivative of $\mu$ with respect to $\nu$ exists.

\end{theorem}
\begin{proof}
For fairly elementary proofs, see \hyperlink{Bartels2003}{Bartels (2003)}.

\end{proof}
(This last theorem is not as general as it could be.)

Note the repetition of `with respect to $\nu$' in various guises; let us fix $\nu$ (assumed to be $\sigma$-finite) and take everything with respect to it. Then it is convenient to treat all measurable functions up to [[almost equality|equality almost everywhere]]; and given any absolutely continuous $\mu$ (also assumed to be $\sigma$-finite), we speak of [[the]] Radon--Nikodym derivative of $\mu$.

\hypertarget{notation}{}\subsection*{{Notation}}\label{notation}

See also the discussion of notation at [[measure space]].

Using the simplest notation for integrals, the definition of Radon--Nikodym derivative reads

\begin{displaymath}
\mu(A) = \int_A f \nu ,
\end{displaymath}
or equivalently

\begin{displaymath}
\int_A \mu = \int_A f \nu .
\end{displaymath}
In other words, the measure $\mu$ is the product of the function $f$ and the measure $\nu$:

\begin{displaymath}
\mu = f \nu ;
\end{displaymath}
and so $f$ is the ratio of $\mu$ to $\nu$:

\begin{displaymath}
f = \mu/\nu .
\end{displaymath}
So this is the simplest notation for the Radon--Nikodym derivative.

However, this notation for integrals is uncommon; one is more likely to see

\begin{displaymath}
\int_A \mathrm{d}\mu = \int_A f \,\mathrm{d}\nu ,
\end{displaymath}
which leads to

\begin{displaymath}
f = \mathrm{d}\mu/\mathrm{d}\nu
\end{displaymath}
for the Radon--Nikodym derivative. But none of these `$\mathrm{d}$'s are really necessary.

We can also use a fuller notation with a dummy variable as the object of the symbol `$\mathrm{d}$':

\begin{displaymath}
\int_{x \in A} \mu(\mathrm{d}x) = \int_{x \in A} f(x) \,\nu(\mathrm{d}x) ;
\end{displaymath}
this leads to

\begin{displaymath}
f(x) = \mu(\mathrm{d}x)/\nu(\mathrm{d}x) ,
\end{displaymath}
which does not give a symbol for $f$ directly. If instead of $\mu(\mathrm{d}x)$ one unwisely writes $\mathrm{d}\mu(x)$, then this gives the previous notation for the Radon--Nikodym derivative.

Now let $\nu$ be [[Lebesgue measure]] on the [[real line]] and let $F$ be an upper [[semicontinuous function]] on the real line, so that $F$ defines a [[Borel measure]] $\mu$ generated by

\begin{displaymath}
\mu({]-\infty,a]}) \coloneqq F(a) .
\end{displaymath}
Then $F$ is [[absolutely continuous function|absolutely continuous]] if and only if $\mu$ is absolutely continuous, in which case the [[derivative]] $F'$ exists almost everywhere and is a Radon--Nikodym derivative of $\mu$. That is,

\begin{displaymath}
\mu/\nu = F' = \mathrm{d}F/\mathrm{d}t .
\end{displaymath}
The presence of `$\mathrm{d}$' on the right-hand side inspires people to put it on the left-hand side as well; but this is spurious, since we really want to write

\begin{displaymath}
\mu = \mathrm{d}F
\end{displaymath}
and

\begin{displaymath}
\nu = \mathrm{d}t ,
\end{displaymath}
where $t$ is the [[identity function]] on the real line.

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

Some fairly elementary proofs prepared for a substitute lecture in [[John Baez]]'s introductory measure theory course are here:

\begin{itemize}%
\item [[Toby Bartels]] (2003): \emph{The Radon Nikodym Theorem}; \href{http://tobybartels.name/notes/#Radon}{web}.

\end{itemize}
The strategy there is based on:

\begin{itemize}%
\item Richard Bradley (1989): An Elementary Treatment of the Radon-Nikodym Derivative, American Mathematical Monthly 96(5), 437--440.

\end{itemize}
[[!redirects Radon-Nikodym derivative]] [[!redirects Radon-Nikodym derivatives]] [[!redirects Radon–Nikodym derivative]] [[!redirects Radon–Nikodym derivatives]] [[!redirects Radon--Nikodym derivative]] [[!redirects Radon--Nikodym derivatives]]

[[!redirects Radon-Nikodym ratio]] [[!redirects Radon-Nikodym ratios]] [[!redirects Radon–Nikodym ratio]] [[!redirects Radon–Nikodym ratios]] [[!redirects Radon--Nikodym ratio]] [[!redirects Radon--Nikodym ratios]]

[[!redirects Radon-Nikodym theorem]] [[!redirects Radon-Nikodym theorems]] [[!redirects Radon–Nikodym theorem]] [[!redirects Radon–Nikodym theorems]] [[!redirects Radon--Nikodym theorem]] [[!redirects Radon--Nikodym theorems]] [[!redirects Radon-Nikodym Theorem]] [[!redirects Radon-Nikodym Theorems]] [[!redirects Radon–Nikodym Theorem]] [[!redirects Radon–Nikodym Theorems]] [[!redirects Radon--Nikodym Theorem]] [[!redirects Radon--Nikodym Theorems]]



\end{document}