Sketch of proof. Use a uniformly distributed real number in [0,1) to define a sequence of coin flips, via its binary digits (with the convention that we do not allow the sequence of binary digits to end with an infinite sequence of 1s). Let A be the subset of [0,1) which defines a sequence that turns the lamp on. Let ρk be the measurable function from [0,1) to [0,1) which flips the kth binary digit where 2−k<m(I)/4. Note that A and ρkA are disjoint by property (2) of our lamp. It is easy to see that we have a violation of the Lebesgue density theorem: any interval in which A has high density, ρkA will have high density in as well for large enough k.
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