Finite and infinite measures

Question

Let $\mathcal{F}$ an algebra of sets and let $\mu_{0}$ a finite and $\sigma$- additive measure.

We denote with $\mathcal{F_\sigma}$ the family all countable unions of sets of $\mathcal{F}$ and with $\mathcal{F_\delta}$ the family all countable intersection of sets of $\mathcal{F}$.

We extend $\mu_{0}$ to two family $\mathcal{F\sigma}$ and $\mathcal{F}_{\delta}$ in the following way:

if $A\in\mathcal{F_{\sigma}}$ \begin{equation} \mu_{1}(A)=\sup\{\mu_{0}(A'), A'\subset A,\; A'\in\mathcal{F}\} \end{equation} else if $B\in\mathcal{F_\delta}$ \begin{equation} \mu_{2}(B)=\inf\{\mu_{0}(B'), B'\supset B,\; B'\in\mathcal{F}\}. \end{equation}

$\mu_1$ and $\mu_2$ can they be $+\infty$?

Thanks!

Answer 1

Obviously $\mu_1(A) \leq \mu(\Omega)$ and $\mu_2(A) \leq \mu(\Omega)$.