I need to apply DCT to bring the limit inside the abslute value...
$\displaystyle \lim_{\tau \rightarrow 0} \left\lbrace \sum_{z \in \mathcal{E}} \mid a_z \mid ^2 \left(\int_{B_{R}} \left|\frac{1}{\sigma(U(z,\tau))}\int_{U(z,\tau) } e^{ ix \cdot \xi}\,d{\sigma \left( \xi\right) }\right|^p \,dx\right) ^\frac{2}{p} \right\rbrace ^\frac{1}{2}$
where $\mathcal{E}$ is a discrete set of points , $B_{R}$ is the ball in $\mathbb{R}^n $ of radius $R$.
$a_z$ are some constants.
$U(z,\tau)$ is a neighborhood of $z$.
Is there any reason why I can't write
$\displaystyle \lim_{\tau \rightarrow 0} \left\lbrace \sum_{z \in \mathcal{E}} \mid a_z \mid ^2 \left(\int_{B_{R}} \left|\frac{1}{\sigma(U(z,\tau))}\int_{U(z,\tau) } e^{ ix \cdot \xi}\,d{\sigma \left( \xi\right) }\right|^p \,dx\right) ^\frac{2}{p} \right\rbrace ^\frac{1}{2} = \left\lbrace \sum_{z \in \mathcal{E}} \mid a_z \mid ^2 \left(\lim_{\tau \rightarrow 0}\int_{B_{R}} \left|\frac{1}{\sigma(U(z,\tau))}\int_{U(z,\tau) } e^{ ix \cdot \xi}\,d{\sigma \left( \xi\right) }\right|^p \,dx\right) ^\frac{2}{p} \right\rbrace ^\frac{1}{2}$
thanks!
What you write is correct provided the discrete set is finite, since $x^{1/2}$ and $x^{2/p}$ are continuous functions. If that set is infinite, then generally no.