p4 - #4. Let * ( E ) be the outer measure dened by formula...

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Math 8601: REAL ANALYSIS. Fall 2010 Homework #4 (due on Wednesday, October 27). 50 points are divided between 5 problems, 10 points each. #1. Let R n be represented in the form R n = S k =1 I k , where { I k } are non-overlapping cubes with edge length 1. Let F k be a closed subset of I k , k = 1 , 2 , ··· . Show that the set F = S k =1 F k is closed. #2. Give an example of families of sets { A α } , { B α } , where α I - an arbitrary set, for which ± ² α A α ³ Δ ± ² α B α ³ " ² α ± A α Δ B α ³ . #3. Let f ( x ) be a strictly positive continuous function on [0 , 1]. Show that the two- dimensional Lebesgue measure of an open set E := { ( x,y ) R 2 : 0 < x < 1 , 0 < y < f ( x ) } coincides with the Riemann integral 1 R 0 f ( x ) dx (see the definition on p.57).
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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 ) &lt; . 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 ) &gt; 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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