Restriction. Let \((X, \mathscr{A}, \mu)\) be a measure space and let \(\mathscr{B} \subset \mathscr{A}\) be a sub- \(\sigma\)-algebra. Denote by \(u:=\left.\muight|_{\mathscr{B}}\) the restriction of \(\mu\) to \(\mathscr{B}\).

(i) Show that \(u\) is again a measure.

(ii) Assume that \(\mu\) is a finite measure [a probability measure]. Is \(u\) still a finite measure [a probability measure]?

(iii) Does \(u\) inherit \(\sigma\)-finiteness from \(\mu\) ?