Non-measurable sets (2). Consider on \(X=\mathbb{R}\) the \(\sigma\)-algebra \(\mathscr{A}:=\left\{A \subset \mathbb{R}: Aight.\) or \(A^{c}\) is countable \} from Example 3.3 (v) and the measure \(\gamma(A)\) from Example 4.5 (ii), which is 0 or 1 according to \(A\) or \(A^{c}\) being countable. Denote by \(\gamma^{*}\) and \(\mathscr{A}^{*}\) the outer measure and \(\sigma\)-algebra which appear in the proof of Theorem 6.1 .

(i) Find \(\gamma^{*}\) if we use \(\mathcal{S}=\mathscr{A}\) in Theorem 6.1 .

(ii) Show that no set \(B \subset \mathbb{R}\), such that both \(B\) and \(B^{c}\) are uncountable, is in \(\mathscr{A}\) or in \(\mathscr{A}^{*}\).

Data from theorem 6.1

Data from example 4.5

Data from example 3.3

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(Carathodory) Let S P(X) be a semi-ring and u: S [0, ] a pre-measure, i.e. a set function with (i) (0) = 0; (ii) (Sn)neNCS, disjoint and S=USES (S) = (Sn). NEN NEN