Real Analysis Homework #4
Due 10/20 1. Let L be the collection of Lebesgue measurable sets of R1 and λ the Lebesgue measure on L. Show that L is invariant under translation and dilation, i.e., if E ∈ L, then E + s ∈ L and rE ∈ L for any s, r ∈ R. Also show that λ(E + s) = λ(E) and λ(rE) = |r|λ(E).
2. Let E be Lebesgue measurable and λ(E) < ∞. Show that for any ε > 0 there exists a finite union of disjoint open intervals I1, · · · , In such that
λ(E4[
In) < ε.
3. Let A the set of numbers in [0, 1] which decimal expansions do not contain digit 5. Show that λ(A) = 0.
4. Given 0 < ε < 1, construct a dense subset E ⊂ [0, 1] such that λ(E) = ε.
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