Consider on \(\mathbb{R}\) the family \(\Sigma\) of all Borel sets which are symmetric w.r.t. the origin. Show that \(\Sigma\) is a \(\sigma\)-algebra. Is it possible to extend a pre-measure \(\mu\) on \(\Sigma\) to a measure on \(\mathscr{B}(\mathbb{R})\) ? If so, is this extension unique? (Problem 9.14 is a continuation of this problem.)

Data from problem 9.14

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(Continuation of Problem 6.3) Consider on R the o-algebra of all Borel sets which are symmetric w.r.t. the origin. Set A+=An [0, 0), A=(-00, 0]n A and consider their symmetrizations A = AU (-A) . Show that for every u M+ (2) with 0