• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
• Functional derivative
• Euler-Lagrange

Radon–Nikodym

A measure $\mu$ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure $\lambda$ (in other words, dominated by $\lambda$ ) if for every measurable set $A$, $\lambda (A)=0$ implies $\mu (A)=0$. This is written as $\mu \ll \lambda$.

If $\nu \ll \mu$, then there is a measurable function $f:X\to [0,\mathrm{\infty })$, such that for any measurable set $A\subseteq X$,

$$\begin{array}{}\text{(1)}& \nu (A)={\int }_{A}fd\mu \end{array}$$

The function  $f$  is called the Radon–Nikodym derivative of $\nu$ with respect to $\mu$ and is denoted by $\frac{d\nu }{d\mu }$.

Riesz–Markov–Kakutani representation theorem

Let $X$ be a locally compact Hausdorff space. For any positive linear functional $\psi$ on $C_c(X)$, the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space $X$, there is a unique regular Borel measure $\mu$ on $X$ such that

$$\begin{array}{}\text{(2)}& \psi (f)={\int }_{X}g(x)d\mu (x)\end{array}$$

for all $g$ in $C_c(X)$.

Functional derivative

Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions etc.), and a functional $F$ defined as

$$\begin{array}{}\text{(3)}& F:M\to \mathbb{R}\text{or}F:M\to \mathbb{C}\end{array}$$

let

$$\begin{array}{}\text{(4)}& \underset{\epsilon \to 0}{lim}\frac{F[\rho +\epsilon g]-F[\rho ]}{\epsilon }& ={[\frac{d}{dϵ}F[\rho +ϵg]]}_{ϵ=0},\end{array}$$

where $g$ is an arbitrary function. This is analogous to the directional derivative

$$\begin{array}{}\text{(5)}& \underset{h\to 0}{lim}\frac{f(\mathbf{x}+h\mathbf{v})-f(\mathbf{x})}{h}={[\frac{d}{d\alpha }f(\mathbf{v}+\alpha \mathbf{u})]}_{\alpha =0}\end{array}$$

The quantity $\epsilon g$ is called the variation of $\rho$. In other words,

$$\begin{array}{}\text{(6)}& \psi :g\mapsto {[\frac{d}{dϵ}F[\rho +ϵg]]}_{ϵ=0}\end{array}$$

is a linear functional, so by the Riesz–Markov–Kakutani representation theorem, this functional is given by integration against some measure. Then $\delta F/\delta \rho (x)$ is defined to be the Radon–Nikodym derivative of this measure and therefore

$$\begin{array}{}\text{(7)}& \psi (g)=\int g(x)\frac{\delta F}{\delta \rho (x)}dx\end{array}$$

We think of the function $\delta F/\delta \rho (x)$ as the gradient of $F$ at the point $\rho$ and

$\int g(x)\frac{\delta F}{\delta \rho (x)}dx$ as the directional derivative at point $\rho$ in the direction of $g$.

Euler-Lagrange

Consider the functional

$$\begin{array}{}\text{(8)}& J[f]={\int }_a^{b}L[x,f(x),f{}^{\prime }(x)]dx\end{array}$$

where $f^{\prime }(x)\equiv df/dx$. If $f$ is varied by adding to it a function $\delta f$, and the resulting integrand $L(x,f+\delta f,f‘+\delta f\prime )$ is expanded in powers of $\delta f$, then the change in the value of $J$ to first order in $\delta f$ can be expressed as follows:

$$\begin{array}{}\text{(9)}& \delta J={\int }_a^b\frac{\delta J}{\delta f(x)}\delta f(x)dx.\end{array}$$

The coefficient of $\delta f(x)$, denoted as $\delta J/\delta f(x)$, is called the functional derivative of $J$ with respect to $f$ at the point $x$. For this example functional, the functional derivative is the left hand side of the Euler-Lagrange equation

$$\begin{array}{}\text{(10)}& \frac{\delta J}{\delta f(x)}=\frac{\partial L}{\partial f}-\frac{d}{dx}\frac{\partial L}{\partial f^{\prime }}\end{array}$$