Existence of a Lebesgue measurable set

[rain hedge]

Let μ denote Lebesgue measure on R
. Show that there exists a Lebesgue measurable set E⊂R
such that for every nonempty open interval J⊂R
we have 0<μ(J∩E)<μ(J).

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[rain hedge]

I tried constructing the set using the rational numbers but couldn't do it.

[zealous jasper]

ofc that wouldn't work, mu(E) > 0 is required

such a borel set E would have to have measure zero

the lebesgue density theorem implies that mu-almost every point in E is a point of density

show that E has no points of density

@rain hedge

[limber tree]

@rain hedge

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