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1. Prove Proposition 1.7 2. Prove that equivalence classes give a series of mutually disjoint sets. Z 1 2 2 |f | < ∞ (Lebesque integral) . 3. Define the set Λ [0, 1] = f : f is measurable and 0 2 (a) Show that Λ [0, 1] is a linear space (you need not repeat proofs of results that were (should have been) taught in a measure theory course). 2 (b) Prove or disprove that for f ∈ Λ [0, 1], if Z 1 f 2 < ∞, then f = 0. 0 (c) Show that M = {f : f = 0 ae} is a subspace of Λ2 [0, 1]. 2 2 2 (d) Define L [0, 1] = Λ [0, 1]/M. For equivalence classes F in L [0, 1], define Z 1 F = 0 some f ∈ F . Show that Z 1 F is well defined, i.e. for F ∈ Λ2 [0, 1] and f, g ∈ F , 0 (e) Prove or disprove that for F ∈ L2 [0, 1], if Z 1 Z 1 f for 0 R1 0 f= R1 0 g. F 2 < ∞, then F = 0. Note For F ∈ L2 [0, 1] 0 and any f ∈ F , F 2 is defined to be the equivalence class associated with f 2 .