Definition 6.1.2 (Outer Integral).label Let $G$ be a locally compact group and $f,\phi\in C_{c}^{+}(G)$ then a covering of $f$ by $\phi$ is any finite sum $\sums{}{j}c_{j}L_{x_j}\phi,c_{j}\geq 0,x_{j}\in G$ such that $f\leq \sums{}{j}c_{j}L_{x_j}\phi$. The outer pseudo integral of $f$ with respect to $\phi$
has the following properties:
- (1)
For each $\al\in (0,1)$, there exists $N_{\al}\in\N$ depending on $f,\phi,\al$ such that $(f:\phi)\leq \frac{N_{\al}\normu{f}}{\al\normu{\phi}}<\infty$
- (2)
$(f:\phi)=(L_{y}f:\phi),\fall y\in G$
- (3)
$(f_{1}+f_{2}:\phi)\leq (f_{1}:\phi)+(f_{2}:\phi)$
- (4)
$(cf:\phi)=c(f:\phi),\fall c>0$
- (5)
$(f_{1}:\phi)\leq (f_{2}:\phi),\fall f_{1}\leq f_{2}$
- (6)
$(f:\phi)\geq \normu{f}/\normu{\phi}$
- (7)
$(f:\phi)\leq (f:\psi)(\psi:\phi),\fall \psi\in C_{c}^{+}(G)$
Proof.
- (1)
Since $\emp\neq U_{\al}\define \curl{x\in G:\phi(x)>\al\supp{\phi}}\suf G$ is open, $\curl{xU_\al:x\in G}$ is an open cover of $G$ by compactness of $\supp{f}$ there exists $x_{1},...,x_{N_\al}$ such that $\supp{f}\suf \cups{N_\al}{j=1}x_{j}U_{\al}$. For each $x\in \supp{f}$ there exists $j$ such that $x\in x_{j}U$ hence $\phi(x_{j}\inv x)>\al\normu{\phi}$. Thus
\begin{align*}\sums{N_\al }{j=1}\frac{\normu{f}}{\al\normu{\phi}}L_{x_j}\phi(x)=\sums{N_\al }{j=1}\frac{\normu{f}}{\al\normu{\phi}}\phi(x_{j}\inv x)> \normu{f}\geq f(x)\end{align*} - (2)
It suffices to prove $(f:\phi)\leq (L_{y}f:\phi)$ since $x\mto L_{x}$ is a group homomorphism. Let $\e>0$ then $L_{y}f\leq \sums{n }{j=1}c_{j}L_{x_j}\phi$ for $c_{1},...,c_{n}$ satisfying $\sums{n }{j=1}c_{j}\leq (L_{y}f:\phi)+\e$ hence $f\leq \sums{n }{j=1}c_{j}L_{y\inv x_j}\phi$.
- (3)
The sum of two coverings is a covering of the sum
- (4)
Covering respects scaling up to the scaling constant
- (5)
By transitivity of $\leq$
- (6)
For any covering we have
\begin{align*}f(x)\leq \sums{}{j}c_{j}L_{x_j}\phi(x)\leq \sums{}{j }c_{j}\normu{\phi}\end{align*}for each $x\in G$ hence taking supremum over $x$ gives the desired result.
- (7)
If $f\leq \sums{}{j }c_{j}L_{x_j}\psi$ and $\psi\leq \sums{}{k }b_{k} L_{y_k}\phi$ then $f\leq \sums{}{j,k }c_{j}b_{k}L_{x_jy_k}\phi$.
$\square$
Post a Comment