User:Boris Tsirelson/sandbox2

here

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

\[\text{ '"`UNIQ-MathJax2-QINU`"' compact}\]

\begin{equation} \mu (B)= \sup \{\mu(K): K\subset B, \text{ '"`UNIQ-MathJax3-QINU`"' compact}\}\, \end{equation}

and having the following property: \begin{equation}\label{e:tight} \mu (B)= \sup \{\mu(K): K\subset B, \mbox{ '"`UNIQ-MathJax4-QINU`"' compact}\}\, \end{equation} (see [Sc]).

The total variation measure of a $\mathbb C$-valued measure is defined on $\mathcal{B}$ as: \[ \abs{\mu}(B) :=\sup\left\{ \sum \abs{\mu(B_i)}: \text{$\{B_i\}\subset\mathcal{B}'"`UNIQ-MathJax7-QINU`"'B$}\right\}. \] 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.