Let \((X, \mathscr{A}, \mu)\) be a measure space which is not \(\sigma\)-finite. Construct an example of a sequence \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{M}(\mathscr{A})\) which converges in measure but whose limit is not unique. Can this happen in a \(\sigma\)-finite measure space?

[ let \(X_{\sigma f}:=\bigcup\{F: \mu(F)<\infty\}\) be the \(\sigma\)-finite part of \(X\). Show that \(X \backslash X_{\sigma f} eq \emptyset\), that every measurable \(E \subset X \backslash X_{\sigma f}\) satisfies \(\mu(E)=\infty\) and that we can change every limit of \(\left(u_{n}ight)_{n \in \mathbb{N}}\) outside \(X_{\sigma f}\).]