Suppose I have a sequence of inner regular measures, is it true that the lim-inf of the sequence an inner regular measure?
I will be more specific for the context of my problem.
Let $\Omega_l$ be such that $\Omega_l \subset \Omega_{l+1}$ and $\cup_l \Omega_l = \Omega$, furthermore $\mu(\Omega_l) , \mu(\Omega) < \infty$, $\mu$ denotes the Lebesgue measure. Let $u_{\varepsilon} \geq 0$ be a sequence of bounded integrable functions, then does $\lim_l \liminf_{\varepsilon} \int_{\Omega_l} u_{\varepsilon}(x)dx = \liminf_{\varepsilon} \int_{\Omega} u_{\varepsilon}(x)dx ?$