User:Boris Tsirelson/sandbox2

the method is as follows: \begin{equation} \begin{gathered} k_1 = h f(x_0,y_0), \quad y_0 = y(x_0),\\ k_2 = h f \big( x_0 + \tfrac13 h, y_0 + \tfrac13 k_1 \big) \, , \\ k_3 = h f \big( x_0 + \tfrac13 h, y_0 + \tfrac16 k_1 + \tfrac16 k_2 \big) \, , \\ k_4 = h f \big( x_0 + \tfrac12 h, y_0 + \tfrac18 k_1 + \tfrac38 k_3 \big) \, , \\ \end{gathered} \end{equation}

The number

the method is as follows:

(2)

The number

here

$$\text{ $K$ compact}$$

\[\text{ '"`UNIQ-MathJax3-QINU`"' compact}\]

\begin{equation} \mu (B)= \sup \{\mu(K): K\subset B, \text{ '"`UNIQ-MathJax4-QINU`"' compact}\}\, \end{equation}

and having the following property: \begin{equation}\label{e:tight} \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ '"`UNIQ-MathJax5-QINU`"' compact}\}\, \end{equation} (see [Sc]).

The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \text{$\{B_i\}\subset\mathcal{B}'"`UNIQ-MathJax8-QINU`"'B$}\right\}. \] In the real-valued case the above definition simplifies as

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and the following identity holds: \begin{equation}\label{e:area_formula} \int_A J f (y) \, dy = \int_{\mathbb R^m} \mathcal{H}^0 (A\cap f^{-1} (\{z\}))\, d\mathcal{H}^n (z)\, . \end{equation}

Cp. with 3.2.2 of [EG]. From \eqref{e:area_formula} it is not difficult to conclude the following generalization (which also goes often under the same name):

\begin{equation}\label{ab} E=mc^2 \end{equation} By \eqref{ab}, it is possible. But see \eqref{ba} below: \begin{equation}\label{ba} E\ne mc^3, \end{equation} which is a pity.