1 answer

Problem 19. Prove that there exists a Jordan measurable set contained in IR which is not...

Question:

Problem 19. Prove that there exists a Jordan measurable set contained in IR which is not a Borel set.

Problem 19. Prove that there exists a Jordan measurable set contained in IR which is not a Borel set.

Answers

Answer: - To Prove that there exist a Jordan measurable set contained in R is not a Borel set. Our goal here is to construct a measurable set which is not Borel set and such a set exists because L measure is completion of Borel measure. We can understand it by example, f: [0, 1] -> [0, 2], f(x) = c(x) + x. f is strictly increasing: f’ = 1 almost everywhere. f is continuous: both c and x are continuous.

F-1 exists: by intermediate value theorem f(2) = 0, f(1) = 2. f-1 is continuous let’s go with claim

fw continus : .umes, dif py..k E:g,re a Eco4 eant.(s).(t.ee))ces 확 1o

.

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