MATH 202 A - Problem Set 13 Walid
نویسنده
چکیده
1. Let z ∈ R, and consider the function fz : R→ R x 7→ x+ z We have fz is Borel-measurable since for all c ∈ R, {x ∈ R|fz(x) ≤ c} = {x ∈ R|x+ z ≤ c} = (−∞, c− z) ∈ B. We also observe that if E ⊆ R, then f−1 z (E) = {x ∈ R|x + z ∈ E} = {−z + (x + z)|x + z ∈ E} = −z + E. Thus we have: if E is a Borel set, then f−1 −y (E) = y + E is a Borel set. Conversely, if y + E is a Borel set, then −y + (y + E) = E is a Borel set. This proves the equivalence. 2. Let E be a Lebesgue measurable set. Then E = E′∆Z for some E′ ∈ B and Z ⊂ Z ′ ∈ B with m(Z ′) = 0. Then we have y + E = f−1 −yE = f−1 −y (E ′∆Z) = f−1 −y (E ′)∆f−1 −y (Z) since inverse images preserve set operations = (y + E′)∆(y + Z) where y +E′ ∈ B, and y + Z ⊆ y + Z ′ ∈ B, and m(y + Z ′) = m(Z ′) = 0. Therefore y +E is Lebesgue measurable. Conversely, if y + E is Lebesgue measurable, then −y + (y + E) = E is Lebesgue measurable. This proves the equivalence.
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