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 .