Q1. Let \(\mu\) and \(\nu\) be finite measures on a measurable space \((\Omega, \mathcal{A})\). Let \(\hat{\mathcal{A}}_{\mu}\) and \(\hat{\mathcal{A}}_{\nu}\) be the completion of the two measures, i.e.,
\(\hat{\mathcal{A}}_{\mu} = \{A: A_1 \subset A \subset A_2 \text{ with } A_1, A_2 \in \mathcal{A} \text{ and } \mu(A_2 \setminus A_1) = 0\}\),
\(\hat{\mathcal{A}}_{\nu} = \{A: A_1 \subset A \subset A_2 \text{ with } A_1, A_2 \in \mathcal{A} \text{ and } \nu(A_2 \setminus A_1) = 0\}\).
(1a). Show by example that \(\hat{\mathcal{A}}_{\mu}\) and \(\hat{\mathcal{A}}_{\nu}\) need not be equal.
(1b). Prove or disprove: \(\hat{\mathcal{A}}_{\mu} = \hat{\mathcal{A}}_{\nu}\) implies \(\mu\) and \(\nu\) have exactly the same sets of measure zero. That is, if the conclusion is correct, prove it. If not, provide an example.
(1c). Prove or disprove: that \(\mu\) and \(\nu\) have exactly the same sets of measure zero implies \(\hat{\mathcal{A}}_{\mu} = \hat{\mathcal{A}}_{\nu}\). That is, if the conclusion is correct, prove it. If not, provide an example.
