Non-measurable sets (1). Let (mu) be a measure on (mathscr{A}={emptyset,[0,1),[1,2),[0,2)}, X=[0,2)), such that (mu([0,1))=mu([1,2))=frac{1}{2}) and (mu([0,2))=1). Denote

Question:

Non-measurable sets (1). Let $\mu$ be a measure on $\mathscr{A}=\{\emptyset,[0,1),[1,2),[0,2)\}, X=[0,2)$, such that $\mu([0,1))=\mu([1,2))=\frac{1}{2}$ and $\mu([0,2))=1$. Denote by $\mu^{*}$ and $\mathscr{A}^{*}$ the outer measure and $\sigma$-algebra which appear in the proof of Theorem 6.1 .

(i) Find $\mu^{*}(a, b)$ and $\mu^{*}\{a\}$ for all \(0 \leqslant a

(ii) Show that $(0,1),\{0\} otin \mathscr{A}^{*}$.

Data from theorem 6.1