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\begin{document}


\begin{center}
\textbf{\huge{Teor\'ia Geom\'etrica de la Medida}}
\end{center}

\begin{center}
 \bf{\LARGE{Ejercicios}}
\end{center}

\begin{description}

\item[Ejercicio 29:] (Clase 17, 21/06)

Probar que en $\R^n$, las medidas de Hausdorff $h^s$, con $0\leq
s<n$, no son de Rad\'on.


\item[Ejercicio 30:] (Clase 17, 21/06)

Sea $O:\R^n\longrightarrow\R^m$ una transformaci\'on ortogonal, es
decir, $$\langle Ox,Oy\rangle=\langle x, y\rangle\quad\forall\,
x,y\in\R^n.$$Si $H^n=\frac{\alpha(n)}{2^n}h^n$, probar que $$H^n(A)=H^n(O(A))\quad\forall\,
A\subseteq\R^n$$ y que
$$dim_H(A)=dim_H(O(A))\quad\forall\,
A\subseteq\R^n.$$ %siendo $H^n=\frac{\alpha(n)}{2^n}h^n$.

\item[Ejercicio 31:] (Clase 17, 21/06)

Sea $L:\R^n\longrightarrow\R^m$ una transformaci\'on lineal. Si se
define $$\nu(A):=H^n(L(A))\quad\forall\,A\subseteq\R^n,$$ demostrar
que:
\begin{description}
\item[i)]$\nu$ es una medida de Rad\'on,
\item[ii)] $\nu$ es absolutamente continua respecto a la medida de
Lebesgue\\
n-dimensional.
\end{description}

\item[Ejercicio 32:] (Clase 17, 21/06)

Sea $A\subseteq\R^n$ medible Lebesgue y sea
$f:\R^n\longrightarrow\R^m$ ($n\leq m$) una funci\'on Lipschitz.\\
Dado $k\in\N$, sea
$$B_k=\Big\{\frac{1}{2^k}\Big([0,1)^n+j\Big) : j\in\Z^n\Big\},$$ el conjunto de los cubos di\'adicos de
nivel $k$. \\Si se definen las funciones
$g_k:\R^m\longrightarrow\R\cup\{+\infty\}$, por $$g_k(y):=\sum_{Q\in
B_k}\chi_{f(A\cap Q)}(y),$$ y $$g(y):=\lim_{k\to +\infty}g_k(y),$$
probar que $$g(y)=H^0(A\cap f^{-1}(\{y\})).$$



\end{description}
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