Consider the functionals defined by $$\Phi_p(u):=\left\lbrace\dfrac{1}{\vert \Omega\vert}\int_{\Omega}\vert u\vert^p \, \mathsf{d}x\right\rbrace^{1/p},$$ for $p\neq 0$. I want check the well known result $$\lim_{p\to 0}\Phi_p(u)=\exp\left(\dfrac{1}{\vert\Omega\vert}\int\log\vert u\vert \, \mathsf{d}x\right).$$ I know that there is an increasing sequence of elementary functions $(u)_n$ such that $\int_\Omega u_n\to\int_\Omega u$. Now by using this the assertion holds true for any $u_n$, i.e.
$$\lim_{p\to 0}\Phi_p(u_n)=\exp\left(\dfrac{1}{\vert\Omega\vert}\int\log\vert u_n\vert \, \mathsf{d}x\right).$$ My problem is the following...
Is the equality $$\lim_{n\to\infty}\lim_{p\to 0}\Phi_p(u_n)=\lim_{p\to 0}\lim_{n\to \infty}\Phi_p(u_n)$$ true? If it is we finished. Can you help me?
