We have a measure space $(X,\mathcal{A}, \mu)$. We are defining $\mathcal{A}_{\mu}$ to be the collection of subsets $A$ of $X$ for which there are sets $E$ and $F$ in $\mathcal{A}$ such that $E \subset A \subset F$ and $\mu(F \setminus E) = 0$ and a function defined as $\overline{\mu}: \mathcal{A}_{\mu} \to [0, \infty ]$ by letting $\overline{\mu}(A)$ be the common value $\mu(E) = \mu(F)$ where $E$ and $F$ belong to $\mathcal{A}$ are the sets associated with $A$ in the definition of $\mathcal{A}_{\mu}$. We have to show:
- $\mathcal{A}_{\mu}$ contains the $\sigma$-field, $\mathcal{A}$
- $\mathcal{A}_{\mu}$ is also a $\sigma$-field.
- $\overline{\mu}$ is a measure on $(X,\mathcal{A}_{\mu})$
We are thinking to prove (1) by taking an arbitrary set $U \in \mathcal{A}$ and we need to show that this belongs to $\mathcal{A}_{\mu}$. Our strategy is to take $E$ and $F$ equal to $U$ (if we ignore the condition of proper subsets) but we are unsure if this is the correct way to go about it. We think this right because we will have to relax the condition of proper subset when proving (2) to show $X \in \mathcal{A}_{\mu}$.
Can anyone confirm if we are doing this correctly. If not, please suggest a correct way.
Note: We have solved (2) and (3) assuming (1) holds. We are only stuck in (1).