Consider the Borel \(\sigma\)-algebra \(\mathscr{B}[0, \infty)\) and write \(\lambda=\left.\lambda^{1}ight|_{[0, \infty)}\) for Lebesgue measure on the half-line \([0, \infty)\).

(i) Show that \(\mathscr{G}:=\{[a, \infty): a \geqslant 0\}\) generates \(\mathscr{B}[0, \infty)\).

(ii) Show that \(\mu(B):=\int_{B} \mathbb{1}_{[2,4]} \lambda(d x)\) and \(ho(B):=\mu(5 \cdot B), B \in \mathscr{B}[0, \infty)\) are measures on \(\mathscr{B}[0, \infty)\) such that \(\left.hoight|_{\mathscr{G}} \leqslant\left.\muight|_{\mathscr{G}}\) but \(ho \leqslant \mu\) fails.

Why does this not contradict Lemma 16.6 ?

Data from lemma 16.6

Lemma 16.6 Let and v be two measures on the measurable space (X, 4) and let & be a semi-ring such that o(8)=A. If p(S)