Problem 3 (Vitali's covering theorem). Let F be any collection of nondegenerate closed balls in R" with sup{diam(B): B € F} <∞. Then there exists a countable family G of disjoint balls in F such that UBCU B. BEG NOTATION: If B B(x, r) is a cosed ball in Rn, we write B concentric closed ball with radius 5 times the radius of B. = BEF = B(x, 5r) to denote the

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I need help with this Theorem (Vitali's Covering Theorem) in trying to prove it. I can only use the notations given. I tried it myself first but I didn't get anywhere with it. Please show all steps and maybe images will help too! 

Problem 3 (Vitali's covering theorem). Let F be any collection of nondegenerate
closed balls in R" with
sup{diam(B): B € F} <∞.
Then there exists a countable family G of disjoint balls in F such that
UBCU B.
BEG
BEF
NOTATION: If B
=
B(x,r) is a cosed ball in R", we write B
concentric closed ball with radius 5 times the radius of B.
-
B(x, 5r) to denote the
Transcribed Image Text:Problem 3 (Vitali's covering theorem). Let F be any collection of nondegenerate closed balls in R" with sup{diam(B): B € F} <∞. Then there exists a countable family G of disjoint balls in F such that UBCU B. BEG BEF NOTATION: If B = B(x,r) is a cosed ball in R", we write B concentric closed ball with radius 5 times the radius of B. - B(x, 5r) to denote the
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