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Unformatted text preview: #4. Let * ( E ) be the outer measure dened by formula (1.12), in which is a premeasure on an algebra A P ( X ) with ( X ) < . Show that for arbitrary subsets E 1 ,E 2 X , we have * ( E 1 E 2 ) + * ( E 1 E 2 ) * ( E 1 ) + * ( E 2 ) . #5. Let f ( x ) be a continuous function on [0 , 1]. Show that the set E = { x [0 , 1] : f ( x ) > f ( y ) for all y [0 ,x ) } is Borel measurable. Hint. Note that the set E remains the same if we replace f ( x ) by f 1 ( x ) := sup y x f ( y ) ....
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 Fall '08
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 Math

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