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  3. problem 4. suppose that a ⊂ r satisfies m1(a

Problem 4. Suppose that A ⊂ R satisfies m1(A) = 0, where m1 denotes the one-dimensional Lebesque measure....

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Advance Math

Problem 4. Suppose that A ⊂R satisfies m1(A) = 0, wherem1 denotes the one-dimensional Lebesquemeasure. Suppose f : R →R2 satisfies

|f(x) − f(y)| ≤ (|x − y|)1/2, forevery x, y ∈ R.

Show that m2(f(A)) = 0,m2 denotes the two-dimensional Lebesque measureon R2 .

Answer & Explanation Solved by verified expert
3.7 Ratings (616 Votes)
I dont exactly remember measure theory having done it manyyears ago but Ill sketch a proof which you can make morerigorousYou can show that fA is measurable following from it beingouter measurable Caratheodory criterion To show that    See Answer
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