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I was leafing through some old exams of our Real analysis course, and I found this puzzling problem:

**"Let A⊂ℝ be Lebesgue-measurable so that for all a∈A, i = 1,2, ...**

(1) m

Claim: m

(1) m

_{1}( {x∈ℝ | a+(3/4)i^{-2}< x < a + i^{-2}} ) < i^{-3}Claim: m

_{1}(A) = 0."Initially I thought this may have something to do with the Lebesgue density theorem that has been used a lot during the course. However, to me it looks like condition (1) doesn't really set any boundaries to what kind of set A could be. (1) only tells us that :

(2) m

_{1}({x∈ℝ | a+(3/4)i

^{-2}< x < a + i

^{-2}}) = lenght(a+(3/4)i

^{-2}, a + i

^{-2}) = |a + i

^{-2}- a+(3/4)i

^{-2}| = 1/4 i

^{-2}.

Now, 1/4 i

^{-2}< i

^{-3}is true for any i =1,2,3. Condition (2) seems to be true for ANY set E⊂ℝ, if i = 1,2, or 3, even those for which m

_{1}(E) >0 (such as an interval). The only condition we get for the set A is that it has to be Lebesgue-measurable.

Any help would be appreciated.